Chapter 3. Straight Lines

Straight lines are the simplest locus which a point can have. We have studied the concept of locus in previous chapter. In this chapter we will study different forms of the straight lines and problems related to these concepts. It goes without saying that it is one of most fundamental and important concepts in coordinate geometry.

We know certain facts about straight lines:

Infinitely many lines can be drawn through a point. See Figure 3.1, “Infinite lines through a point.”
Figure 3.1. Infinite lines through a point.
Infinitely many lines can be drawn in a direction. See Figure 3.2, “Infinite parallel lines.”
Figure 3.2. Infinite parallel lines.
One and only one line can be drawn through a fixed point in a given direction. See Figure 3.3, “A line through a point having direction.”
Figure 3.3. A line through a point having direction.
One and only one line can be drawn through two given points. See Figure 3.4, “A line through two points.”
Figure 3.4. A line through two points.

3.1. Angle of Inclination of a Line

The angle of elevation of any line which cuts the 𝑥-axis the angle which it makes with x-axis in the positive anti-clockwise direction.

Figure 3.5. Inclination angle of a straight line.

3.2. Slope or Gradient of a Line

If the angle of inclination is $θ$ then $t a n θ$ is called the slope or gradient of the straight line. Usually it is denoted by $𝑚$ . Thus, the slope of the lines shown in Figure 3.5, “Inclination angle of a straight line.” are $\tan 45^{\circ}$ and $\tan 135^{\circ}$ i.e. $1$ and $- 1$ .

If a line passes through two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ then the slope is given by $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ as we will see soon.

3.3. Equation of a Straight Line

Equation of a straight line is the equation in $x$ and $y$ , which is satisfied by the coordinates of all the points on the line and is not satisfied by any point which is not on the line. It is always an equation of first degree in $x$ and $y$ . Thus, the general form would be $a x + b y + c = 0$ , where $a, b$ and $c$ are constants. There are other forms or representations of this equation which will follow this section. The equation may be an equation in $x$ and $y$ or only $x$ or only $𝑦$ .

3.4. Equation of Lines Parallel to Axes

3.4.1. Equation of $x$ -axis

Let $P (x, y)$ be an arbitrary point on $x$ -axis. Since $P (x, y)$ lies on $x$ -axis, therefore, $y = 0$ . The relation $y = 0$ is true for all points on $x$ -axis, while $x$ varies. Hence the equation of $x$ -axis is $y = 0$ (3.1)

3.4.2. Equation of $y$ -axis

Let $P (x, y)$ be an arbitrary point on $y$ -axis. Since $P (x, y)$ lies on $y$ -axis, therefore, $x = 0$ . The relation $x = 0$ is true for all points on $y$ -axis, while $y$ varies. Hence the equation of $y$ -axis is $x = 0$ (3.2)

3.4.3. Equations of Lines Parallel to $y$ -axis

Figure 3.6. Lines parallel to $y$ -axis.

Lines parallel to <math>
<mi>y</mi>
</math>-axis.

Let $A B$ be a straight line parallel to $y$ -axis at a distance $a$ from it on the positive side of $x$ -axis.

Let $A B$ meets $x$ -axis at $L$ , then $O L = a$ . Let $P (x, y)$ be any point on the line $A B$ . Now $x = O L$ or $x = a$ . This relation is satisfied for all points on the line $A B$ and is not satisfied by any point, which does not lie on $A B$ .

Hence, the equation of the straight line parallel to $y$ -axis at a distance $a$ from it on the positive side of $x$ -axis is $x = a$ (3.3) Similarly if the line is on negative side of $x$ -axis is $x = - a$ (3.4)

3.4.4. Equations of Lines Parallel to $y$ -axis

Proceeding like previous section, the equation of the straight line parallel to $x$ -axis at a distance $b$ ffom it on the positive side of $y$ -axis is $y = b$ (3.5) and if on negative side is $y = - b$ (3.6)

3.5. Forms of Straight Line

3.5.1. Slope-Intercept Form

We will find an equation to a straight line whose slope is $m$ and which cuts an intercept $c$ on the $y$ -axis i.e. which passes through the point $(0, c)$ .

Figure 3.7. Slope intercept form of a straight line.

Let $A B$ be a line whose slope is $m$ and which cuts an intercept $c$ on $y$ -axis. Let $∠ B H X = θ$ and line $A B$ cuts $y$ -axis at $Q$ .

Then from the figure $O Q = c$ and $\tan θ = m$ .

Let $P (x, y)$ be any point on the line $A B$ . From $P$ we draw $P L ⊥ Q X$ and from $Q$ we draw $Q M ⊥ P L$ .

Since $∠ P H L = θ ∴ P Q M = θ$ .

Now in $△ P Q M, \tan θ = \frac{P M}{Q M} = \frac{P L - M L}{Q M} = \frac{P L - Q O}{O L} = \frac{y - c}{x} = m$ $y = m x + c$ (3.7)

Note: If the line passes through origin, then $c = 0$ and the equation of the line becomes $y = m x$ .

3.5.2. Point-Slope Form

We will find an equation to a straight line whose slope is $m$ and which passes through a point $(x_{1}, y_{1})$ . See Figure 3.8, “Point slope form of a straight line.”.

Let $A B$ be a straight line whose slope is $m$ and which passes through the point $Q (x_{1}, y_{1})$ . Let the line $A B$ cut the $x$ -axis at $H$ and $∠ B H X = θ$ then $\tan θ = m$ .

Let $P (x, y)$ be any point on line $A B$ . From $P$ and $Q$ we draw $P L$ and $Q M$ perpendiculars on $O X$ and from $Q$ we draw $Q N$ perpendicular on $P L$ .

Since $P = (x, y) ∴ O L = x$ and $P L = y$ and since $Q = (x_{1}, y_{1}) ∴ O M = x_{1}$ and $Q M = y_{1}$ .

Since $∠ B H X = θ ∴ ∠ P Q N = θ$

Now from $△ P Q N, \tan θ = \frac{P N}{Q N} = \frac{y - y_{1}}{x - x_{1}} ∴ m = \frac{y - y_{1}}{x - x_{1}}$ $y - y_{1} = m (x - x_{1})$ (3.8)

Figure 3.8. Point slope form of a straight line.

Aliter: Since $m$ is the slope of the line therefore its equation may be written as $y = m x + c$ . Since $(x_{1}, y_{1})$ lies on this line therefore we can write $y_{1} = m x_{1} + c \Rightarrow c = y_{1} - m x_{1}$ . Thus, $y - y_{1} = m (x - x_{1})$ .

3.5.3. Two-Point Form

We will find an equation of a straight line which passes through two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . See Figure 3.9, “Two-point form of a straight line.”.

Let $A B$ be a line which passes through two points $Q (x_{1}, y_{1})$ and $R (x_{2}, y_{2})$ . Let $P (x, y)$ be any point on $A B$ .

From $P, Q$ and $R$ we drop $P L, Q M$ and $R N$ perpendiculars to x-axis. From $Q$ we drop perpendicular $Q H$ to $P L$ and from $R$ we drop perpendicular $R K$ to $Q M$ .

$∵ P = (x, y) ∴ O L = x, P L = y, Q = (x_{1}, y_{1}) ∴ O M = x_{1}, Q M = y_{1}$ and $R = (x_{2}, y_{2}) ∴ O N = x_{2}, R N = y_{2}$ .

From similar $△ P H Q$ and $△ Q K R$ , we have

$\frac{P H}{Q K} = \frac{H Q}{K R} ∴ \frac{y - y_{1}}{y_{1} - y_{2}} = \frac{x_{1} - x}{x_{2} - x_{1}}$ $y - - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ (3.9)

Figure 3.9. Two-point form of a straight line.

Aliter: Since points $P (x, y), Q (x_{1}, y_{1})$ and $R (x_{2}, y_{2})$ are collinear, therefore $| \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{1} & 1 \end{matrix} | = 0 \Rightarrow y - - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ .

Aliter: Let the equation of straight line be $y = m x + c$ . Since it passes through $(x_{1}, y_{1})$ and $(x - 2, y_{2})$ , therefore $y_{1} = m x_{1} + c, y_{2} = m x_{2} + c \Rightarrow m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Now $c = y_{1} - m x_{1} = y_{1} - \frac{y - y_{1}}{x_{2} - x_{1}} x_{1}$

Substituting we get $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ .

3.5.4. Intercept Form

We will find the equation of the straight line which cuts the intercept $a$ and $b$ at $x$ -axis and $y$ -axis respectively. See Figure Figure 3.10, “Intercept form of a straight line.”

Let $A B$ be a line which cuts intercept $a$ and $b$ on $x$ -axis and $𝐴 𝐵$ -axis respectively. Let line $A B$ cut $x$ -axis at $Q$ and $y$ -axis at $R$ , then $O Q = a$ and $O R = b$ .

Let $P (x, y)$ be any point on line $A B$ , then $O L = x$ and $P L = y$ .

Now in similar $△ Q L P$ and $△ O Q R$ , we have

Figure 3.10. Intercept form of a straight line.

$\frac{Q L}{O Q} = \frac{P L}{O R} \Rightarrow \frac{a - x}{b} = \frac{y}{b}$ $\frac{x}{a} + \frac{y}{b} = 1$ (3.10)

Aliter: Since $O Q = a$ and $O R = b$ , therefore $Q = (a, 0)$ and $R = (0, b)$ . Since points $P (x, y), Q (a, 0)$ and $R (0, b)$ are collinear, therefore, $| \begin{matrix} x & y & 1 \\ a & 0 & 1 \\ 0 & b & 1 \end{matrix} | = 0 \Rightarrow b x + a y = a b \Rightarrow \frac{x}{a} + \frac{y}{b} = 1$

Aliter: $Δ Q O R = Δ P O Q + Δ P R O \Rightarrow \frac{1}{2} a b = \frac{1}{2} a y + \frac{1}{2} b x \Rightarrow \frac{x}{a} + \frac{y}{b} = 1$ .

3.5.5. Normal Form

We will find the equation of the straight line upon which the length of the perpendicular from the origin is $p$ and this normal makes an angle 𝛼 with the positive direction of $x$ -axis.

Let $A B$ be a line and $O P$ be perpendicular from the origin to $A B$ . Let $O P = p$ and $∠ P O X = α$ (the point can be different than $P$ but in this diagram $P$ is such that $O P$ is perpendicular to $A B$ ).

Let $P (x, y)$ be any point on the line $A B$ . From $P$ we draw $P L$ perpendicular to $O X$ . Let line $A B$ cut the $x$ -axis and $y$ -axis at $Q$ and $R$ respectively.

Figure 3.11. Normal form of a straight line.

Since $P = (x, y) ∴ O L = x, P L = y$ . Since $∠ P O Q = α ∴ ∠ P Q O = 90^{\circ} - α, ∠ L P Q = α$

From $△ P L Q, L Q = P L \tan α = y \tan α$ . Now $O P = O Q \cos α = (O L + L Q) \cos α = (x + y \tan α) \cos α$ $x \cos α + y \sin α = p$ (3.12)

Aliter: $O Q = p \sec α$ and since $∠ O R P = α, O R = p \csc α$

Hence, $Q = (p \sec α, 0), R = (0, p \csc α)$

Now the points $P (x, y), Q (p \sec α, 0)$ and $R (0, p \csc α)$ are collinear, therefore, $| \begin{matrix} x & y & 1 \\ p \sec α & 0 & 1 \\ 0 & p \csc α & 1 \end{matrix} | = 0 \Rightarrow x (- p \csc α) - y p \sec α + 1. p^{2} \sec α \csc α = 0$

$\Rightarrow x \cos α + y \sin α = p$ .

3.5.6. Distance Form or Parametric Form

We will find the equation of the straight line passing through the point $(x_{1}, y - 1)$ and making an angle $θ$ with the positive direction of $x$ -axis.

Let $A B$ be a line which passes through the point $Q (x_{1}, y_{1})$ and makes an angle $θ$ with the positive direction of $x$ -axis.

Let $P (x, y)$ be a point on the line $A B$ . From $P$ and $Q$ we draw $P L$ and $Q M$ perpendicular on $x$ -axis. From $Q$ , we draw $Q K$ perpendicular to $P L$ .

Figure 3.12. Distance form of a straight line.

$∵ P = (x, y) ∴ O L = x, P L = y ∵ Q = (x_{1}, y_{1}) ∴ O M = x_{1}, Q M = y_{!}$ .

$∵ ∠ B H X = θ ∴ ∠ B H O = π - θ ∴ ∠ P Q K = π - θ$

Let $P Q = r$ . From $△ P K Q, \cos (π - θ) = \frac{K Q}{P Q} \Rightarrow - \cos θ = \frac{x - x_{1}}{r} \Rightarrow \cos θ = \frac{x - x_{1}}{r} \Rightarrow \frac{x - x_{1}}{\cos θ} = r$

Also, $\sin (π - θ) = \frac{P K}{P Q} \Rightarrow \sin θ = \frac{y - y_{1}}{r} \Rightarrow \frac{y - y_{1}}{\sin θ = r}$ . Thus, $\frac{x - x_{1}}{\cos θ} = \frac{y - y_{1}}{\sin θ} = r$ (3.13)

Corollary: From Eq. 3.13 we can say that $x = x_{1} + r \sin θ, y = y_{1} + r \cos θ$ (3.14)

which is parametric form of a straight line.

If $Q (x_{1}, y_{1})$ be a point on a line $A B$ , which makes an angle $θ$ with the positive direction of $x$ -axis, then there will be two points on line $A B$ and their coordinates will be $(x_{1} + r \cos θ, y_{1} + r \sin θ)$ and $(x_{1} - r \cos θ, y_{1} - r \sin θ)$ . These two points will be on two opposite sides of $Q$ on the line $A B$ .

3.5.7. General Euqation of a Straight Line

We will show that the general equation of first degree in $x$ and $y$ always represent a straight line.

Let the general equation of first degree in $x$ and $y$ be $a x + b y + c = 0$ .

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ be two points on this line. Then $a x_{1} + b y_{1} + c = 0$ and $a x_{2} + b y_{2} + c_{2} = 0$ .

Multiplying the two equations by $m$ and $n$ respectively and adding gives us $a (m x_{1} + n x_{2}) + 𝑏 (m y_{1} + n y_{2}) + c (m + n) = 0$ .

$\Rightarrow a \frac{m x_{1} + n x_{2}}{m + n} + b \frac{m y_{1} + n y_{2}}{m + a} + c = 0$ .

Thus, point $(\frac{m x_{1} + n x_{2}}{m + n}, \frac{m y_{1} + n y_{2}}{m + n})$ will lie on the straight line we have chosen.

Since $m$ and $n$ are arbitrary numbers, therefore, each point on the line $A B$ will lie on this locus. Hence, the general equation of a straight line is $a x + b y + c = 0$ (3.15)

Aliter: Let the general equation of first degree in $x$ and $y$ be $a x + b y + c = 0$ .

Let $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ be any three points on the above line. Then $a x_{1} + b y_{1} + c = 0, a x_{2} + b y_{2} + c = 0$ and $a x_{3} + b y_{3} + c = 0$ .

$\Rightarrow | \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$

Thus, $A, B, C$ are collinear. Thus, the equation $a x + b y + c = 0$ is equation of a straight line.

Converse: Every straight line may be represented by a first degree equation in $x$ and $y$ . We know that through two points one and only one straight line can be drawn. Let $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ be two points on the straight line and let $P (x, y)$ be any point on it. Now since $P, A, B$ are collinear

$\Rightarrow | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | = 0 \Rightarrow (y_{1} - y_{2}) x - (x_{1} - x_{2}) y + x_{1} y_{2} - x_{2} y_{1} = 0$ ,

which is of the form $a x + b y + c = 0$ , where $a = y_{1} - y_{2}, b = x_{2} - x_{1}$ and $c = x_{1} y_{2} - x_{2} y_{1}$ .

Since the equation $a x + b y + c = 0$ is satisfied by all points on the line $A B$ and it will not be satisfied by the coordinates of any point which does not lie on the line $A B$ , hence, it represents the equation of line $A B$ .

3.6. Representing General Equation in Standard Forms

3.6.1. Slope Intercept Form

We will reduce the equation $A x + B y + C = 0$ to $y = m x + c$ .

Given equation is $A x + B y + c = 0 \Rightarrow y = - \frac{A}{B} x - \frac{C}{B}$ , which is of the form $y = m x + c$

Comparing we have $m = - \frac{A}{B}$ and $c = - \frac{C}{B}$ .

3.6.2. Intercept Form

We will reduce the equation $A x + B y + C = 0$ to $\frac{x}{a} + \frac{y}{b} = 1$ .

This reduction is possible only if $C \neq 0$ .

Given equation is $A x + B y + C = 0 \Rightarrow - \frac{A}{C} x - \frac{B}{C} y = 1$ , where $C \neq 0$ .

$\Rightarrow \frac{x}{- \frac{C}{A}} + \frac{y}{- \frac{C}{B}} = 1$ , which is of the form $\frac{x}{a} + \frac{y}{b} = 1$ , where $a = - \frac{C}{A}$ and $b = - \frac{C}{B}$ .

3.6.3. Normal Form

We will reduce the equation $A x + B y + C = 0$ to the form $x \cos α + y \sin α = p$ , where $α$ is the angle made by the perpendicular on the line from origin and $p$ is the length of the perpendicular.

Given equation is $A x + B y = - C$

Case I: When $C < 0$ i.e. $- C > 0$ .

Dividing both sides by $\sqrt{A^{2} + B^{2}}$ gives us

$\frac{A}{\sqrt{A^{2} + B^{2}}} + \frac{B}{\sqrt{A^{2} + B^{2}}} = - \frac{C}{\sqrt{A^{2} + B^{2}}}$ , which is of the form $x \cos α + y \sin α = p$ .

Case II: When $C > 0$ i.e. $- C < 0$

Given equation can be written as $- A x - B y = C$

$\Rightarrow - \frac{A}{\sqrt{A^{2} + B^{2}}} - \frac{B}{\sqrt{A^{2} + B^{2}}} = \frac{C}{\sqrt{A^{2} + B^{2}}}$ , which is of the form $x \cos α + y \sin α = p$ .

3.7. Angle between Two Straight Lines

Figure 3.13. Angle between two straight lines.

We will find the angle between two straight lines given by the equations $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ .

Let $A B$ and $C D$ be two given lines whose equations are $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ .

Let $A B$ and $C D$ cut each other at $P$ and they cut the $x$ -axis at points $L$ and $M$ respectively.

Let $∠ M P L = ψ$ and $∠ A L X = α, ∠ D M X = β$ . Now since line $A B$ makes an angle $α$ with $x$ -axis and its slope is $m_{1}$ . $∴ m_{1} = \tan α$ .

Again since $C D$ makes an angle $β$ with $x$ -axis and its slope be $m_{2}$ . $∴ m_{2} = \tan β$ .

From the figure $α = ψ + β \Rightarrow ψ = α - β$

$\tan ψ = \tan (α - b e t a) = \frac{\tan α \tan β}{1 + \tan α + \tan β} = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}$

If $∠ P L D = ϕ$ then $ψ + ϕ = π \Rightarrow ϕ = π - ψ$

$∴ \tan ϕ = - \tan ψ = - \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}$

Since $ψ$ and $ϕ$ are the two angles between the lines $A B$ and $C d$ , it follows that the angle $θ$ between these two lines is given by

$\tan θ = \pm \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} \Rightarrow θ = \tan^{- 1} | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |, π - \tan^{- 1} | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$

If $θ$ is the acute angle between the lines then $\tan θ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$ (3.16)

Notes:

here are two angles between two non-perpendicular lines. One of them is acute and the other one is obtuse and their sum is $180^{\circ}$ . Thus, if acute angle $θ$ between the two lines is known then the other angle will be $180^{\circ} - θ$ .
The formula $\tan θ = \pm \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}$ is valid only when $m_{1}$ and $m_{2}$ are defined.
f both the lines are perpendicular to 𝑥-axis then the angle between them is $0^{\circ}$ .
If one of the lines is perpendicular to $x$ -axis and the other line makes an angle of $θ$ with positive direction of $x$ -axis then the angle between them is $| 90^{\circ} - θ |$ .

3.7.1. Parallelism and Perpendicularity

Our equation for angle between two straight lines is $\tan θ = \pm \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}$ . If the lines are parallel then the value of $θ$ would be $0$ . Thus, $m_{1} = m_{2}$ . For special case when two lines are parallel to $y$ -axis i.e. $m_{1} = m_{2} = \infty$ then both the lines are taken as parallel.

For lines to be perpendicular then $\tan θ = \infty \Rightarrow 1 + m_{1} m_{2} = 0 \Rightarrow m_{1} m_{2} = - 1$ . When either of $m_{1}$ or $m_{2}$ is not defined then one of them has to be parallel to $x$ -axis while the other has to be parallel to $y$ -axis.

3.7.2. To find the angle between straight lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$

Writing the equation in slope intercept form we find that the slopes are given by $m_{1} = - \frac{a_{1}}{b_{1}}$ and $m_{2} = - \frac{a_{2}}{b - 2}$ .

If $θ$ be the angle between the two lines then $\tan θ = \pm \frac{- \frac{a_{1}}{b_{2}} + \frac{a_{2}}{b_{1}}}{1 + \frac{a_{1} a_{2}}{b_{1} b_{2}}} = \pm \frac{a_{2} b_{1} - a_{1} b_{2}}{a_{1} a_{2} + b_{1} b_{2}}$

$θ = \tan^{- 1} | \frac{a_{2} b_{1} - a_{1} b_{2}}{a_{1} a_{2} + b_{1} b_{2}} |$ or $θ = π - \tan^{- 1} | \frac{a_{2} b_{1} - a_{1} b_{2}}{a_{1} a_{2} + b_{1} b_{2}} |$

Clearly, the lines will be parallel if $θ = 0$ i.e. $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$ (3.17) and the lines will be perpendicular if $a_{1} a_{2} + b_{1} b_{2} = 0$ (3.18)

3.7.3. Lines Parallel to Another Line

We will find equation of a line parallel to another line in standard form i.e. $a x + b y + c = 0$ .

Let the line parallel to give line is $l x + m y + n = 0$ .

From Eq. 3.17 these two lines will be parallel if $\frac{l}{a} = \frac{m}{b} = p$ (say) then $l = a p, m = b p$ , which makes the equation of the line as $a x + b y + \frac{n}{p} = 0$ or $a x + b y + k = 0$ , where $k = \frac{n}{p}$ .

Thus, equation of any line parallel to $a x + b y + c = 0$ is given by $a x + b y + k = 0$ (3.19)

3.7.4. Lines Perpendicular to Another Line

We will find equation of a line perpendicular to another line in standard form i.e. $a x + b y + c = 0$ .

Let the line perpendicular to give line is $l x + m y + n = 0$ .

From Eq. 3.18 these two lines will be parallel if $a l + b m = 0$ or $\frac{l}{b} = - \frac{m}{a} = p$ (say)

Thus, $b p x - a p y + n = 0 \Rightarrow b x - a y + \frac{n}{p} = 0 \Rightarrow b x - a y + k = 0$ , where $k = \frac{n}{p}$

Thus, equation of any line perpendicular to $a x + b y + c = 0$ is given by $b x - a y + k = 0$ (3.20)

3.8. Point of Intersection

We will find the point of intersection of two lines in standard form i.e. between the lines given by $a_{1} x + b_{y} + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ .

Let $P (α, β)$ is the point of intersection of these two lines then $a_{1} α + b_{1} β + c_{1} = 0$ and $a_{2} α + b_{2} β + c_{2} = 0$ .

By cross-multiplication we have $\frac{α}{b_{1} c_{2} - b_{2} c_{1}} = \frac{β}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

Thus, the point of intersection is given by $(\frac{b_{1} c_{2} - b_{2} c_{1}}{a_{1} b_{2} - a_{2} b_{1}}, \frac{c_{1} a_{2} - c_{2} a_{1}}{a_{1} b_{2} - a_{2} b_{1}})$ .

As you can see in case of parallel lines $a_{1} b_{2} - a_{2} b_{1} = 0$ and then $α$ and $β$ are not defined.

In this case neither $c_{1} a_{2} - c_{2} a_{1}$ nor $b_{1} c_{2} - b_{2} c_{1}$ will be zero, otherwise $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ i.e. the two lines will be coincident or same.

3.8.1. Line Passing through Point of Intersection of Two Lines

We will find the general equation of a straight line passing through point of intersection of two straight lines in standard form i.e. lines given by $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ .

We consider the straight line in first degree equation $a_{1} x + b_{1} y + c_{1} + k (a_{2} x + b_{2} y + c_{2}) = 0$ .

Let $P (α, b e t a)$ be the point of intersection then $a_{1} α + b_{1} β + c_{1} = 0$ and $a_{2} α + b_{2} β + c_{2} = 0$ and $a_{1} α + b_{1} β + c_{1} + k (a_{2} α + b_{2} β + c_{1}) = 0$ .

Clearly, the third equation is an equation of a line passing through $P$ . Thus, our desired equation is $a_{1} x + b_{1} y + c_{1} + k (a_{2} x + b_{2} y + c_{2}) = 0$ (3.21)

3.9. Concurrency of Three Straight Lines

We will find condition for concurrency of three lines in standard form i.e. the lines are given by $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ .

Let $P (α, b e t a)$ be the point where these three lines meet. Then $P$ will lie on all three lines. Thus, $\frac{α}{b_{1} c_{2} - b_{2} c_{1}} = \frac{β}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$

Thus, the point of intersection is given by $(\frac{b_{1} c_{2} - b_{2} c_{1}}{a_{1} b_{2} - a_{2} b_{1}}, \frac{c_{1} a_{2} - c_{2} a_{1}}{a_{1} b_{2} - a_{2} b_{1}})$ .

Now this point will lie on the third line and therefore will satisfy the equation for third line. Hence, $a_{3} \frac{b_{1} c_{2} - b_{2} c_{1}}{a_{1} b_{2} - a_{2} b_{1}} + b_{3} \frac{c_{1} a_{2} - c_{2} a_{1}}{a_{1} b_{2} - a_{2} b_{1}} + c_{2} = 0$ .

Thus, the required condition for concurrency of three lines is give by $a_{3} (b_{1} c_{2} - b_{2} c_{1}) + b_{3} (c_{1} a_{2} - c_{2} a_{1}) + c_{3} (a_{1} b_{2} - a_{2} b_{1}) = 0$ (3.22)

Aliter: Equations of given lines are $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ .

Let $P (α, b e t a)$ be the point where these three lines meet. Then $P$ will lie on all three lines. Thus, $a_{1} α + b_{1} β + c_{1} = 0, a_{2} α + b_{2} β + c_{2} = 0$ and $a_{3} α + b_{3} β + c_{3} = 0$ .

Eliminating $α, b e t a$ we can write that $| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | = 0$ , which expands to Eq. 3.22.

Corollary: The straight lines $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ . will be concurrent if and only if there exists three constants $l, m, n$ (not all zero at the same time) such that $l (a_{1} x + b_{1} y + c_{1}) + m (a_{2} x + b_{2} y + c_{2}) + n (a_{3} x + b_{3} y + c_{3}) = 0$ identically i.e. for all values of $x$ and $y$ .

Since $l, m, n$ are not zero at the same time let $n \neq 0$ . Let $(α, b e t a)$ be the point of intersection of first two lines. Then

$a_{1} α + b_{1} β + c_{1} = 0$ and $a_{2} α + b_{2} β + c_{2} = 0$

Since the given condition is true for all $x$ and $y$ , therefore $l (a_{1} α + b_{1} β + c_{1}) + m (a_{2} α + b_{2} β + c_{2}) + n (a_{3} α + b_{3} β + c_{3}) = 0$

$\Rightarrow l .0 + m .0 + n (a_{3} α + b_{3} β + c_{3}) = 0 \Rightarrow a_{3} α + b_{3} β + c_{3} = 0$

Thus, the third line also passes through $(α, β)$ .

3.10. Two Sides of a Straight Line

Every line divides the plane in two regions. Any point which does not lie on the line can be only on one side of the straight line.

We will show that a point $(α, β)$ will be on one or the other side of the line $a x + b y + c = 0$ according as the expression $a α + b β + c > 0$ or $< 0$ .

Figure 3.14. Side of a point.

Let 𝐴𝐵 be the given line whose equation is $a x + b y + c = 0$ . Let $P (α, β)$ be a point which does not lie on this line.

From $P$ we draw $P Q$ perpendicular on $x$ -axis. Let $P Q$ cut line $A B$ at $R$ . Clearly, $x$ coordinate of $P$ and $R$ are same.

Let $R = (α_{1}, β_{1})$ . Since it lies on the line, therefore, $a α_{1} + b β_{1} + c_{1} = 0$ .

When $b > 0$ , on left side of the Figure 3.14, “Side of a point.” $P Q > R Q ∴ β > β_{1}$ or $b β > b β_{1}$ or $a α + b β + c > a α_{1} + b β_{1} + c$ or $a α + b β + c > 0$ .

Similarly, for right side of the figure we can establish $a α + b β + c < 0$ .

When $b < 0$ , on left side of the Figure 3.14, “Side of a point.” $P Q > R Q ∴ β > β_{1}$ or $b β < b β_{2}$ or $a α + b β + c < a α + b β_{1} + c$ or $a α + b β + c < 0$ .

Similarly, for right side of the figure we can establish $a α + b β + c > 0$ .

Thus, we see that $a α + b β + c > 0$ or $< 0$ according as the point $P (α, b e t a)$ lies on one or the other side of the line $a x + b y + c = 0$ .

Corollary: It follows from previous article that two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ will lie on the same side or opposite side of the line $a x + b y + c = 0$ according as $a x_{1} + b y_{1} + c$ and $a x_{2} + b y_{2} + c$ are of the same sign or opposite sign.

3.11. Length of a Perpendicular

We will find the length of the perpendicular from the point $(α, β)$ to the line $a x + b y + c = 0$ .

Figure 3.15. Length of a perpendicular.

Let the given line be $a x + b y + c = 0$ and given point be $P (α, b e t a)$ . We have to find the length of the perpendicular from the point $P$ on line $A B$ .

We draw $P M$ perpendicular to $A B$ and join $P A$ and $P B$ . Let $P M = p$ .

Putting y = 0 in the equation for the given line gives us $a x + c = 0 \Rightarrow x = - \frac{c}{a} ∴ A = (- \frac{c}{a}, 0)$ .

Putting $x = 0$ in the equation for the given line gives us $b y + c = 0 \Rightarrow y = - \frac{c}{b} ∴ B = (0, - \frac{c}{b})$ .

Now $Δ P A B = \frac{1}{2} | [α (0 + \frac{c}{b}) + - \frac{c}{a} (- \frac{c}{b} - β) + 0 (β - 0)] | = \frac{1}{2} | \frac{c}{a b} | | a α + b β + c |$

Again $Δ P A B = \frac{1}{2} . A B . P M = \frac{1}{2} \sqrt{{(- \frac{c}{a} - 0)}^{2} + {(0 + \frac{c}{b})}^{2}} . p = \frac{1}{2} | \frac{c}{a b} | \sqrt{a^{2} + b^{2}} . p$

Equating two obtained equations for $Δ P A B$ gives us the length of the perpendicular, which is $p = \frac{| a α + b β + c |}{\sqrt{a^{2} + b^{2}}}$ (3.23)

Aliter: Let $P M$ make an angle $θ$ with the positive direction of $x$ -axis, then the equation of $P M$ in distance form is $\frac{x - α}{\cos θ} = \frac{y - β}{\sin θ} = r$

Coordinates of any point on $P M$ at a distance $r$ from $P (α, β)$ will be $(α + r \cos θ, β + r \sin θ)$ .

Since $P M = p$ , therefore coordinates of $M$ is $(α + p \cos θ + β + p \sin θ)$ , which will lie on the line $a x + b y + c = 0$ , thus,

$a (α + p \cos θ) + b (β + p \sin θ) + c = 0 \Rightarrow p (a \cos θ + b \sin θ) = - (a α + b β + c)$

Now slope of $A B = - \frac{a}{b} ∴$ slope of $P M = \frac{b}{a} ∴ \tan θ = \frac{b}{a}$ or $\frac{a}{\cos θ} = \frac{b}{\sin θ} = k$ (say) $∴ a = k \cos θ, b = k \sin θ$

$\Rightarrow a^{2} + b^{2} = k^{2} ∴ k = \pm \sqrt{a^{2} + b^{2}} ∴ a = \pm \sqrt{a^{2} + b^{2}} \cos θ \Rightarrow \cos θ = \pm \frac{a}{\sqrt{a^{2} + b^{2}}}$ . Similarly $\sin θ = \pm \frac{b}{\sqrt{a^{2} + b^{2}}}$

Putting the values of $\cos θ$ and $\sin θ$ in the equation $p (\cos θ + \sin θ) = - (a α + b β + c)$ gives us

$p = \mp \frac{a α + b β + c}{\sqrt{a^{2} + b^{2}}}$

Since $p$ is positive, therefore, $p = \frac{| a α + b β + c |}{\sqrt{a^{2} + b^{2}}}$ (3.24)

Aliter(Calculus Method):

Figure 3.16. Length of a perpendicular.

Let $Q (x, y)$ be any point on the line $A B$ . Now $P Q$ will be the length of the perpendicular if $P Q$ is minimum. Hence, length of the perpendicular from $P$ on $A B$ will be the least value of $P Q$ when the point $Q (x, y)$ varies.

Let $P Q^{2} = z$ . Also, $P Q$ will be least if and only if $P Q^{2}$ i.e. $z$ is least.

Now $z = (x - α)^{2} + (y - β)^{2}$

Since $Q (x, y)$ lies on the line $a x + b y + c = 0$ , therefore, $y = - \frac{a x + c}{b}$

$\Rightarrow z = (x - α)^{2} + {(- \frac{a x + c}{b} - β)}^{2}$

$\frac{d z}{d x} = 2 (x - α) + \frac{1}{b^{2}} .2 (a x + b β + c)$

For maxima and minima of $z$ , $\frac{d z}{d x} = 0 \Rightarrow \frac{y - β}{x - α} . - \frac{a}{b} = - 1$

$∴$ Slope of line $P Q$ . Slope of line $A B = - 1$

Thus, when $\frac{d z}{d z} = 0, P Q ⊥ A B$ .

Since maximum length of $P Q$ is not possible as it will be $\infty$ (tends to $\infty$ ), and hence, $\frac{d z}{d z} = 0$ gives the minimum length of $P Q$ .

Length of perpendicular, $\sqrt{z} = \sqrt{(x - α)^{2} + (y - β)^{2}}$

But if $\frac{d z}{d z} = 0$ , then $\frac{x - α}{a} = \frac{y - β}{b} = (a (x - α) + \frac{b (y - β)}{a^{2} + b^{2}}) = - \frac{a α + b β + c}{a^{2} + b^{2}}$

$\Rightarrow z = p = \frac{| a α + b β + c |}{\sqrt{a^{2} + b^{2}}}$

Note: Length of perpendicular from origin on line $a x + b y + c = 0$ is $\frac{| c |}{\sqrt{a^{2} + b^{2}}}$ .

3.12. Bisectors of Angles between Straight Lines

Figure 3.17. Bisectors of angles between straight lines.

We will find the equation of the bisectors of the angles between the straight lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ .

Let the given lines be $A B$ and $C D$ whose equations are $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ .

Let $L M$ and $R S$ be the two bisectors of the angles between $A B$ and $C D$ . Let $P (x, y)$ be the point on any bisector. Since $P$ lies on a bisector, therefore, the lengths of perpendiculars on two lines will be equal.

Thus, the length of perpendicular from $P$ to $A B$ will be equal to the length of the perpendicular from $P$ to $C D$ .

$\Rightarrow \frac{| a_{1} x + b_{1} y + c_{1} |}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{| a_{2} x + b_{2} y + c_{2} |}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ . Thus,

$\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \pm \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ (3.25) are the equations of two bisectors.

Note: If $P (x, y)$ is taken on the bisector of the angle which contains the origin then either $O (0, 0)$ and $P (x, y)$ will lie on the same sides of two lines. Thus,

$a_{1} x + b_{1} y + c_{1} > 0$ and $a_{2} x + b_{2} y + c_{2} > 0$ or $O (0, 0)$ and $P (x, y)$ will lie on the opposite side of the two lines i.e. $a_{1} x + b_{1} y + c_{1} < 0$ and $a_{2} x + b_{2} y + c_{2} < 0$

Then equation of bisectors will be $\frac{| a_{1} x + b_{y} + c |}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{| a_{2} x + b_{2} y + c_{2} |}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

i.e. $\frac{a_{1} x + b_{y} + c}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ , when both $a_{1} x + b_{1} y + c_{1}$ and $a_{2} x + b_{2} y + c_{2}$ are positive or $- \frac{a_{1} x + b_{y} + c}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = - \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ , when both $a_{1} x + b_{1} y + c_{1}$ and $a_{2} x + b_{2} y + c_{2}$ are negative.

Thus, in both the cases equation of the bisector containing the origin when $c_{1}$ and $c_{2}$ are both positive is

$\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ (3.26)

When both $c_{1}$ and $c_{2}$ are positive, then the equation of the bisector of the angle between the lines which does not contain the origin is $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = - \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ (3.27) The two bisectors are perpendicular to each other.

3.12.1. Finding Bisector of the Acute and Obtuse Angles

To find the bisector of the acute and obtuse angles take any line out of $a_{1} x + b_{y} + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ and any of the bisectors obtained. Let 𝜃 be the angle between them. Find $| \tan θ |$ . The bisector considered will be the bisector of the acute angle or obtuse angle between the lines according as $θ < 45^{\circ}$ or $θ > 45^{\circ}$ i.e. according as $| \tan θ | < 1$ or $> 1$ .

Aliter: Let the equations of the two lines be $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ , where $c_{1} > 0$ and $c_{2} > 0$ .

Then the equation $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ is the equation of the bisector of the acute or obtuse angle between the lines according as $a_{1} a_{2} + b_{1} b_{2} < 0$ or $> 0$ .

Slope of bisector $= m_{1} = - \frac{a_{1} \sqrt{a_{2}^{2} + b_{2}^{2}} - a_{2} \sqrt{a_{1}^{2} + b_{1}^{2}}}{b_{1} \sqrt{a_{2}^{2} + b_{2} 62} - b_{2} \sqrt{a_{1}^{2} + b_{1}^{2}}}$ and slope of first line is $m_{2} = - \frac{a_{1}}{b_{1}}$

Angle between first line and bisector is $\tan θ = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} =$ $\frac{a_{1} b_{2} - a_{2} b_{1}}{\sqrt{(a_{1} b_{2} - a_{2} b_{1})^{2} + (a_{1} a_{2} + b_{1} b_{2})^{2} - (a_{1} a_{2} + b_{1} b_{2})}}$

Let $α = a_{1} b_{2} - a_{2} b_{1}$ and $β = a_{1} a_{2} + b_{1} b_{2}$ then $| \tan θ | = \frac{| α |}{\sqrt{α^{2} + β^{2} - β}}$

If $β < 0$ and $\sqrt{α^{2} + β^{2} - β} > | α | - β > | α |$ , therefore, $| \tan θ | < 1$ , and hence, the bisector is the bisector of acute angle between the lines.

If $β > 0$ and $\sqrt{α^{2} + β^{2}} \leq | α | + | β | \Rightarrow \sqrt{α^{2} + β^{2} - β} \leq | α |$ , therefore, $| \tan θ | > 1$ , and hence, the bisector is the bisector of obtuse angle between the lines.

Similarly, we can show that the equation $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = - \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ is the equation of the bisector of the acute or obtuse angles according as $a_{1} a_{2} + b_{1} b_{2} > 0$ or $< 0$ .

3.12.2. Bisectors Between Lines Containing a Given Point

Figure 3.18. Bisectors of angles between straight lines.

Let the equations of the lines be $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ .

Let $(α, β)$ be a given point. If $a_{1} α + b_{1} β + c_{1}$ and $a_{2} α + b_{2} β + c_{2}$ are of the same sign then equation of the bisector of the angle containing the point $(α, β)$ is $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ (3.28)

Let $(x_{1}, y_{1})$ be a point on the bisector then $\frac{a_{1} x_{1} + b_{1} y_{1} + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{a_{2} x_{1} + b_{2} y_{1} + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

Since $\sqrt{a_{1}^{2} + b_{1}^{2}}$ and $\sqrt{a_{2}^{2} + b_{2}^{2}}$ are both posiitive, therefore, $a_{1} x_{1} + b_{1} y_{1} + c_{1}$ and $a_{2} x_{1} + b_{2} y_{1} + c_{2}$ are of the same sign.

Case I: When both $a_{1} x_{1} + b_{1} y_{1} + c_{1}$ and $a_{2} x_{1} + b_{2} y_{1} + c_{1}$ are positive.

Since $a_{1} α + b_{1} β + c_{1}$ and $a_{1} x_{1} + b_{1} y_{1} + c_{1}$ are both positive, therefore, points $P (x_{1}, y_{1})$ and $Q (α, b e t a)$ will lie on the same side of the line $a_{1} x + b_{1} y + c_{1} = 0$ . Again since $a_{2} α + b_{2} β + c_{1}$ and $a_{2} x_{1} + b_{2} y_{1} + c_{2}$ are both posiitive, therefore, points $P (x_{1}, y_{1})$ and $Q (α, b e t a)$ will lie on the same side of the line $a_{2} x + b_{2} y + c_{2} = 0$ .

The figure will be like Figure 3.18, “Bisectors of angles between straight lines.”

Case II: When both $a_{1} x_{1} + b_{1} y_{1} + c_{1}$ and $a_{2} x_{1} + b_{2} y_{1} + c_{1}$ are negative.

Figure 3.19. Bisectors of angles between straight lines.

In this case $a_{1} α + b_{1} β + c_{1}$ and $a_{1} x_{1} + b_{1} y - 1 + c_{1}$ are of opposite sign, therefore, points $P (x_{1}, y_{1})$ and $Q (α, β)$ will lie on opposite side of the line $a_{1} x + b - 1 y + c_{1} = 0$ .

Again since $a - 2 α + b_{2} β + c_{2}$ and $a_{2} x_{1} + b_{2} y_{1} + c_{2}$ are of opposite sign, therefore, points $P (x_{1}, y_{1})$ and $Q (α, β)$ will lie on opposite side of the line $a_{2} x + b_{2} y + c_{1} = 0$ . Thus, in this case figure will like Figure 3.19, “Bisectors of angles between straight lines.”

Similarly, if $a_{1} α + b_{1} β + c_{1}$ and $a_{2} α + b_{2} β + c_{2}$ are of opposite sign then the equation of the bisector of the angle containing the point $(α, β)$ is $\frac{a_{1} x + b_{1} y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = - \frac{a_{2} x + b_{2} y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$ (3.29)

Aliter: Let the equations of lines $A B$ and $C D$ are $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ , where $c_{1}$ and $c_{2}$ are positive.

These equations in normal form will be

$- \frac{a_{1} x - b_{1} y}{\sqrt{a_{1}^{2} + b_{1}^{2}}} = \frac{c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$ and $- \frac{a_{2} x - b_{2} y}{\sqrt{a_{2}^{2} + b_{2}^{2}}} = \frac{c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$

Let $\cos α = - \frac{a_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}, \cos β = - \frac{a_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}, \sin α = - \frac{b_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$ , and $\sin β = - \frac{b_{2}}{a_{2}^{2} + b_{2}^{2}}$

Now $\cos (β - α) = \cos α \cos β + \sin α \sin β = \frac{a_{1} a_{2} + b_{1} b_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2}}}$

$β - α$ will be acute or obtuse according as $a_{1} a_{2} + b_{1} b_{2} > 0$ or $< 0$ .

Now $∠ C A B$ will be acute or obtuse according as $β - α$ is obtuse or acute.

But $∠ C A B$ contains the origin, therefore, origin will be contained in the acute or obtuse angle according as $a_{1} a_{2} + b_{1} b_{1} < 0$ or $> 0$ .

Hence, bisector of the angle between the lines will be bisector of the acute or obtuse angle according as origin lies in the acute or obtuse angle.

3.13. Problems

Find the equation of the straight line cutting off an intercept $5$ from the positive direction of $y$ -axis, and inclined at angle $45^{\circ}$ to the $x$ -axis.
Find the equation of the straight line passing through the point $(2, - 3)$ and cutting off intercepts, equal but of opposite signs from the two axes.
Find the equation of the straight line which passes through the point $(- 5, 4)$ and is such that the portion of it between the axes is doivided by the point in the ratio of $1 : 2$ .
Find the normal form of the equation $x + y \sqrt{3} + 7 = 0$ .
Find the equation of the straight line which passes through the points $(- 1, 3)$ and $(4, - 2)$ .
Find the equation of the straight line cutting off intercept unity from the positive direction of the $y$ -axis and inclines at $45^{\circ}$ to the $x$ -axis.
Find the equation of the straight line cutting off intercept $- 5$ from the $y$ -axis and being equally inclined to the axes.
Find the equation of the straight line cutting off intercept $2$ from the negative direction of $y$ -axis and inclined at $30^{\circ}$ to $O X$ .
Find the equation of the straight line cutting off intercept $- 3$ from the $y$ -axis and inclined at an angle $\tan^{- 1} \frac{3}{5}$ to the $y$ -axis.
Find the equation of the straight line cutting off intercepts $3$ and $2$ from the axes.
Find the equation of the straight line cutting off intercepts $- 5$ and $6$ from the axes.
Find the equation of the straight line which passes through the point $(5, 6)$ and has intercept on the axes equal in magnitude and both positive. Find the equation if intercepts are equal in magnitude but opposite in sign.
Find the equations of the straight lines which passes through the point $(1, - 2)$ and cut off equal distances from the two axes.
Find the equation of the straight line which passes through the point $(x^{'}, y^{'})$ and is such that the given point bisects the part intercepted between the axes.
Find the equation of the straight line which passes through the point $(- 4, 3)$ and is such that the portion of it between the axes is divided by the point in the ratio $5 : 3$ .
Find the equation of the straight line passing through the points $(0, 0)$ and $(2, - 2)$ .
Find the equation of the straight line passing through the points $(3, 4)$ and $(5, 6)$ .
Find the equation of the straight line passing through the points $(- 1, 3)$ and $(6, - 7)$ .
Find the equation of the straight line passing through the points $(0, - a)$ and $(b, 0)$ .
Find the equation of the straight line passing through the points $(a, b)$ and $(a + b, a - b)$ .
Find the equation of the straight line passing through the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ .
Find the equation of the straight line passing through the points $(a t_{1}, \frac{a}{t_{1}})$ and $(a t_{2}, \frac{a}{t_{2}})$ .
Find the equation of the straight line passing through the points $(a \cos ϕ_{1}, a \sin ϕ_{1})$ and $(a \cos ϕ_{2}, a \sin ϕ_{2})$ .
Find the equation of the straight line passing through the points $(a \cos ϕ_{1}, b \sin ϕ_{1})$ and $(a \cos ϕ_{2}, b \sin ϕ_{2})$ .
Find the equation of the straight line passing through the points $(a \sec ϕ_{1}, b \tan ϕ_{1})$ and $(a \sec ϕ_{2}, b \tan ϕ_{2})$ .
Find the equations of the sides of the triangle the coordinates of whose angular points are respectively $(1, 4), (2, - 3)$ and $(- 1, - 2)$ .
Find the equations of the sides of the triangle the coordinates of whose angular points are respectively $(0, 1), (2, 0)$ and $(- 1, - 2)$ .
Find the equations of the diagonals of the rectangle the equations of whose sides are $x = a, x = a^{'}, y = b$ and $y = b^{'}$ .
Find the equation of the straight line which bisects the distance between the points $(a, b)$ and $(a^{'}, b^{'})$ and also bisects the distance between the points $(- a, b)$ and $(a^{'}, - b^{'})$ .
Find the equations of the straight lines which go through the origin and trisect the portion of the straight line $3 x + y = 12$ which is intercepted between the axes of coordinates.
Find the equation of the straight line which makes an angle of $15^{\circ}$ with the positive direction of $x$ -axis, and which cuts an intercept of length $4$ on the negative direction of $y$ -axis.
Find the equation of the straight line which cuts off an intercept $4$ from the $x$ -axis and makes an angle of $30^{\circ}$ with the $y$ -axis.
Find the equation of the straight line which passes through the point $(1, 2)$ and makes an angle $θ$ with the positive direction of 𝑥-axis where $\cos θ = - \frac{1}{3}$ .
Find the equation of the line joining the points $(- 1, 3)$ and $(4, - 2)$ .
A line through the point $A (2, 0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$ -axis is rotated about $A$ in clockwise direction through an angle of $15^{\circ}$ . Find the equation of the straight line in the new position.
Find the equation of the internal bisector of the $∠ B A C$ of the $△ A B C$ whose vertices $A, B, C$ are $(5, 2), (2, 3), (6, 5)$ respectively.
A rectangle has two opposite vertices at the point $(1, 2)$ and $(5, 5)$ . If the other vertices lies on the line $x = 3$ , find the equation of the sides of the triangle.
In the given figure $P Q R$ is an equilateral triangle and $O S P T$ is a square. If $O T = 2 \sqrt{2}$ units, find the equation of lines $O T, O S, S P, Q R, P R$ and $P Q$ .
Figure 3.20.
If $D, E, F$ are three points on the sides $B C, A C$ and $A B$ of a $△ A B C$ such that $A D, B E$ and $C F$ are concurrent, then show thta $B D . C E . A F - D C . E A . F B$ .
Find the coordinates of the vertices of a square inscribed in a triangle with vertices $A (0, 0), B (2, 1)$ and $C (3, 0)$ ; given that two of its vertices are on the side $A C$ .
Transform equation $\sqrt{3} y - 3 x = 3$ to the slope intercept form and also find the angle, which this straight line makes with the $x$ -axis.
Find the equation of the straight line which cuts of an intercept of $7$ on $y$ -axis and has the slope $3$ .
Find the equation of the line, which makes an angle of $75^{\circ}$ with $x$ -axis and cuts an intercept of length $3$ on the positive direction of $y$ -axis.
Find the equation of the straight lines which cuts off an intercept $- 5$ from the $y$ -axis and makes an angle of $\sin^{- 1} \frac{12}{13}$ with the $x$ -axis.
Find the equation of the straight line which is parallel to $x$ -axis at a distance of $5$ units from it.
Find the equation of the straight line which is parallel to $y$ -axis at a distance of $4$ units from it towards negative side of $x$ -axis.
Find the equation of the straight lines which pass through $(5, 3)$ and are respectively parallel and perpendicular to the $x$ -axis.
Find the equation of the straight line which intercepts a length of $2$ on the positive direction of $x$ -axis and is inclined at $135^{\circ}$ with the positive direction of $y$ -axis.
Find the equation of a straight line which cuts off an intercept $4$ on the $x$ -axis and has the slope $2$ .
Find the equation of the straight line passing through $(3, - 2)$ and making an angle of $60^{\circ}$ with the positive direction of $y$ -axis.
Find the slope of the line passing through the points $(3, 4)$ and $(1, 2)$ . Also find its equation.
Find the equation of the straight line passing through the points $(a, b)$ and $(a + r \cos θ, b + r \sin θ)$ .
If $(x, y)$ is a point on the straight line joining $(1, - 3)$ and $(- 4, 2)$ show that $x + y + 2 = 0$ .
Prove that the points $(1, 4), (3, - 2)$ and $(- 3, 16)$ are collinear. Also, find the the equation of this straight line.
If the points $(a, b), (a_{1}, b_{1})$ and $(a - a_{1}, b - b_{1})$ are collinear, show that the line joining them passes through the origin.
Find the value of $t$ for which $(1, 2), (- 3, 0)$ and $(t - 1, 3)$ will be collinear. Also find the equation of the straight line.
Show that the straight line passing through the points $(p, q + r)$ and $(q, r + p)$ also passes through the point $(r, p + q)$ .
Find the equation of the straight line passes through the point which divides the line segment joining the points $(- 1, 2)$ and $(4, - 5)$ externally in the ratio $2 : 3$ and the point $(1, 2)$ .
Find the equation of the side $B C$ of the $△ A B C$ whose vertices $A, B, C$ are $(- 1, - 2), (0, 1), (2, 0)$ respectively. Also find the equation of the median through $(- 1, - 2)$ .
The vertices of a triangle are $(1, 2), (2, 3)$ and $(5, 4)$ . Find the equation of its medians.
In what ratio does the line $x + y + 1 = 0$ divide the line segment joining the points $(2, 3)$ and $(- 1, 4)$ ?
Find the ratio in which the line segment joining $(2, 3)$ and $(4, 1)$ is divided by the line joining $(1, 2)$ and $(4, 3)$ .
The vertices of a $△ A B C$ are $(2, 2), (- 1, - 2)$ and $(1, - 3)$ . $D$ is a point on $B C$ such that $B D : D C = 2 : 1$ . Find the ratio in which $A D$ divides the median through $B$ .
For the straight line $y - \sqrt{3} x = 3$ , find the intercept on $y$ -axis and also the angle which the straight line makes with the $x$ -axis.
Find the equation of the straight line which passes through the point $(3, 4)$ and whose intercept on $y$ -axis is twice that on $x$ -axis.
Find the equation of the straight line which passes through the point $(3, 4)$ and is such that the portion of it intercepted between the axes is divided by the point in the ratio $2 : 3$ .
Find the equation of the straight line whose intercepts on $x$ -axis and $y$ -axis are respectively twice and thrice of those by the line $3 x + 4 y = 12$ .
Find the equation of the straight line passing through the origin and the middle point of the intercept of the straight line $a x + b y + c = 0$ between the axes.
Find the equation of the straight lines which pass through the origin and trisect the intercept of the line $3 x + 4 y = 12$ between the axes.
A straight line cuts intercepts from the axes of coordinates the sum of the reciprocal of which is a constant. Show that it always passes through a fixed point.
If the equal sides $A B$ and $A C$ of a right angled isosceles triqangle be produced to $P$ and $Q$ so that $B O . C Q = A B^{2}$ , show that $P Q$ always passes through a fixed point.
Through the point $P (α, β)$ , where $α β > 0$ the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is drawn so as to form with coordinate axes a triangle of area $S$ . If $a b > 0$ , find the least value of $S$ .
Find the equation of the straight line upon which the length of perpendicular from origin is $3 \sqrt{2}$ units and this perpendicular makes an angle of $75^{\circ}$ with the positive direction of $x$ -axis.
Find the equation of the straight line upon which the length of the perpendicular from the origin is $2$ and the slope of the perpendicular is $\frac{5}{12}$ .
A canal is $4 \frac{1}{2}$ kms from a place and the shortest route from this place to canal is exactly north-east. A village is $3$ kms north and $4$ kms east from the place. Does it lie on the canal?
Find the equation of the straight lines which makes a triangle of area $96 \sqrt{3}$ with the axes and perpendicular from the origin to it makes an angle of $30^{\circ}$ with $y$ -axis.
In the given figure $A B C$ is a right-angled iscosceles triangle and $B C D E$ is a square. If $O C = 2$ , find the equation of the sides $A B$ and $B C$ of $△ A B C$ and side $D E$ of the square.
Figure 3.21.
Find the coordinates of the point where the line $\sqrt{3} x + y - 8 = 0$ meets the coordinate axes and also find the length of the perpendicular from the origin upon this line and the angle which this perpendicular makes with the $x$ -axis.
Find the equations of the straight lines which pass through the point $(3, 2)$ and cut off intercepts $a$ and $b$ on the $x$ and $y$ -axes such that $a - b = 2$ .
Find the equation of the line which passes through $P (1, - 7)$ and meets the axes at $A$ and $B$ respectively such that $4 A P - 3 B P = 0$ , where $O$ is the origin.
Find the equation of the straight line which passes through the point $P (2, 6)$ and cuts the axes at the points $A$ and $B$ respectively such that $\frac{A P}{B P} = \frac{2}{3}$ .
Find the equation of the straight line whose intercepts on the axes are twice the intercepts of the line $3 x + 4 y = 6$ .
Find the equation of the straight line passing through $(2, 1)$ and bisecting the portion of the straight line $3 x - 5 y = 15$ lying between the axes.
Find the equations of the straight lines which pass through the origin and trisect the portion of the straight line $2 x + 3 y = 6$ which is intercepted between the axes.
Prove that the points $(5, 1), (11, 4)$ and $(1, - 1)$ lie on a straight line and find its intercepts on the axes and between the axes.
Find the intercepts on the axes of the straight line passing through the points $(1, - 3)$ and $(4, 5)$ .
The lnegth of the perpendicular from the origin to a line is $7$ and the line makes an angle of $150^{\circ}$ with the positive direction of $y$ -axis. Find the equation of the line.
Find the equation of the straight line upon which the length of the perpendicular from the origin is $2$ and this perpendicular makes an angle of $30^{\circ}$ with the positive direction of $y$ -axis.
Find the equation of the line which is at a distance $y$ from the origin and the perpendicular from the origin to the line makes an angle of $60^{\circ}$ with the positive direction of $x$ -axis.
Find the equation of the straight line upon which the length of the perpendicular from the origin is $6$ and the gradient of the perpendicular is $\frac{3}{4}$ .
Find the equation of the line joining the points $(1, 2)$ and $(- 3, 1)$ . Find its intercepts on the axes. If $p$ be the length of the perpendicular from the origin to the line, find the value of $p$ .
Find the equation of the straight line which passes through the point $(3, 2)$ and whose gradient is $\frac{3}{4}$ . Find the coordinates of the points on the line that are $5$ units away from the point $(3, 2)$ .
Find the direction in which a straight line must be drawn through the point $(1, 2)$ so that its point of intersection wth the line $x + y = 4$ is at a distance $\sqrt{\frac{2}{3}}$ from the point $(1, 2)$ .
f the straight line drawn through the point $P (\sqrt{3}, 2)$ makes an angle $\frac{π}{6}$ with the $x$ -axis meets the line $\sqrt{3} x - 4 y + 8 = 0$ at $Q$ . Find the length of $P Q$ .
Find the coordinates of the points at a distance $4 \sqrt{2}$ units from the point $(2, 3)$ in the direction making an angle of $46^{\circ}$ with the positive direction of $x$ -axis.
A line joining two points $A (2, 0)$ and $B (3, 1)$ is rotated about $A$ in anticlockwise direction through an angle $15^{\circ}$ . Find the equation of the line in new position. If $B$ goes to $C$ in the new position what will be the coordinates of $C$ ?
The extremeties of the diagonal of a sqaure are $(1, 1), (- 2, - 1)$ . Obtain the other two vertices and the equation of the other diagonal.
Show that if any line through the variable point $A (k + 1, 2 k)$ meets the line $7 x + y - 16 = 0, 5 x - y - 8 = 0, x - 5 y + 8 = 0$ at $B, C, D$ respectively $A C, A A B$ and $A D$ are in H.P.
The center of a square is at the origin and one vertex is $A (2, 1)$ . Find the coordinates of other vertices of the square.
Show that if $A (x_{1}, y_{1}), B (x_{2}, y_{2}), c (x_{3}, y_{3})$ are the vertices of a triangle, then the equation of the internal bisector of angle $A$ is given by $b | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | + c | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$ , where $b = A C$ and $c = A B$ .
Find the coordinate of the point at a distance $6$ units from the point $(1, 1)$ in the direction making an angle of $60^{\circ}$ with the positive direction of $x$ -axis.
Find the distance of the line $2 x + y = 3$ from the point $(- 1, 3)$ in the direction whose slope is one.
The straight line through $P (x_{1}, y_{1})$ inclined at an angle $θ$ with $x$ -axis meets the line $a x + b y + c = 0$ in $Q$ . Find the length of $P Q$ .
A line through the point $A (2, 0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$ -axis is rotated about $A$ in anticlockwise direction through an angle $15^{\circ}$ . Find the equation of the straight line in the new position.
The straight line $2 x - y = 5$ turns about the point on it where the ordinate is equal to the abcissa through an angle of $45^{\circ}$ in the anti-clockwise direction. Find the equation of the line in the new position.
The straight line $x + 2 y = 4$ is translated parallel to itself by $3$ units in the sense of increasing $x$ and is then rotated by $30^{\circ}$ in the clockwise direction about the point where the shifted straight line cuts the $x$ -axis. Find the equation of the new straight line in the new position.
$A B$ is a side of regular hexagon $A B C D E F$ and is of length a with $A$ as the origin and $A B$ and $A E$ as the $x$ -axis and $y$ -axis respectively. Find the equation of lines $A C, A F$ and $B E$ .
A straight road is at a distance of $5 \sqrt{2}$ miles from a place. The shortest distance of the road from the place is in $N E$ direction. Do the following villages which (i) is $6$ miles East and 4 miles and 4 miles North from the place, lie on the road or not. (ii) is $4$ miles East and $3$ miles north from the place, lie on the road or not?
A straight line $x - y + 1 = 0$ cuts the $y$ -axis at $A$ . This line is rotated about $A$ in the clockwise direction by $75^{\circ}$ . Fine the equation of the new straight line.
Find the equations of all sides of the isosceles $△ A B C$ and the side $B E$ and $C D$ of the square $B C D E$ in the figure where $O C = 2$ units.
Figure 3.22.
The mid-point of the line segment joining $(3, - 1)$ and $(1, 1)$ is shifted by two units(in the sense of increasing $y$ ) perpendicular to the line segment. Find the coordinates of the point in the new position.
The point $(1, 1)$ is translated parallel to the line $2 x = y$ in the first quadrant through a unit distance. Find the new position of the point.
The point $A (2, - 1)$ is translated parallel to the line $x - y = 3$ by a distance of $4$ units. If the new position $A^{'}$ of the point is in the (i) first quadrant (ii) third quadrant then find $A^{'}$ .
Two particles start from the same point $(2, - 1)$ , one moving $2$ units along the line $x + y = 1$ and the other $5$ units along the line $x - 2 y = 4$ . If the particles move towards increasing $y$ , find their new positions and the distance between them.
One end of a thin elastic straight string is fixed at $A (4, - 1)$ and the other end $B$ is at $(1, 2)$ in the unstretched condition. If the string is stretched to triple its length, find the coordinates of the other end in this stretched position.
The line 𝐴𝐵 whose equation is $x - y = 2$ cuts the $x$ -axis at $A$ and $B$ is $(4, 2)$ . The line $A B$ is rotated about $A$ through $45^{\circ}$ in the anticlockwise direction. Find the new position of $B$ and the equation of the line in new position.
Let $x y$ plane be vertical. A particle dropped gently from $(- 1, 1)$ in the plane rebounds on the floor returns to $\frac{2}{3}$ rd of the height from which it has fallen. The equation of the line of intersection of $x y$ plane and the floor is $x + 2 y = 3$ . Find the highest position of the particle after one rebound.
A line is drawn through $A (4, - 1)$ parallel to the line $3 x - 4 y + 1 = 0$ . Find the coordinates of the two points on this line which are at a distance of $5$ units from $A$ .
Find the distance of the point $(3, 5)$ from the line $2 x + 3 y = 14$ measured parallel to the line $x - 2 y = 1$ .
Find the distance of the point $(2, 5)$ from the line $3 x + y + 4 = 0$ measured parallel to the line $3 x - 4 y + 8 = 0$ .
The point $(1, 3)$ and $(5, 1)$ are two opposite sides of a rectangle and the other two vertices lie on the line $y = 2 x + c$ . Find $𝑐$ and other vertices.
A line is drawn from $(x^{'}, y^{'})$ in the direction $α$ with the $x$ -axis, to meet $A x + B y + C = 0$ . Prove that the lengeth is $| \frac{A x^{'} + B y^{'} + c}{A \cos α + B \sin α} |$ .
Find the equation of the line passing through the point $P (1, 2)$ cutting the lines $x + y - 5 = 0$ and $2 x - y = 7$ at $A$ and $B$ respectively such that the H.M. of $P A$ and $P B$ is $10$ . Given that $A, B$ lie on the same side of $P$ .
A straight line through the point $A (- 2, 3)$ cuts the line $x + 3 y = 9$ and $x + y + 1 = 0$ at $B$ and $C$ respectively. Find the equation of the line if $A B . A C = 20$ .
A line which makes an acute angle $θ$ with the positive direction of $x$ -axis is drawn through the point $P (3, 4)$ to cut the curve $y^{2} = 4 x$ at $Q$ and $R$ . Show that the lengths of the segments $P Q$ and $P R$ are numerical values of roots of the equation $r^{2} \sin^{2} θ + 4 r (\sin θ - \cos θ) + 4 = 0$ .
Show that if $A (x_{1}, y_{1}), 𝐵 (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ be the vertices of a triangle then the equation of the median through $A$ is given by $b | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | + | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$ .
Find the angle between the lines $x - 2 y + 3 = 0$ and $3 x + y - 1 = 0$ .
Find the angle between the lines $x + y = 3$ and the line joining the points $(1, 1)$ and $(- 3, 4)$ .
Find the value of $k$ so that the straight line $2 x + 3 y + 4 + 𝑘 (6 x - y + 12) = 0$ is perpendicular to the line $7 x + 5 y - 4 = 0$ .
Prove that the line joining the middle points of the two sides of a triangle is parallel to the third side.
Find the values of $x$ and $y$ for which $A (2, 0), B (0, 2), C (0, 7)$ and $D (x, y)$ are the vertices of an isosceles trapezium in which $A B ∥ C D$ .
Prove that the straight lines $(a + b) x + (a - b) y - 2 a b = 0, (a - b) x + (a + b) y - 2 a b = 0$ and $x + y = 0$ form an isosceles triangle whose vertical angle is $2 \tan^{- 1} \frac{a}{b}$ .
Find the angle between the lines $x = a$ and $b y + c = 0$ .
Find the tangent of the angle between the lines which have intercepts of $3, 4$ and $1, 9$ on $x$ and $y$ axes respectively.
Prove that the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{b} - \frac{y}{a} = 1$ are perpendicular to each other.
Show that the line joining $(2, - 3)$ and $(- 1, 2)$ is perpendicular to the line joining $(3, 7)$ and $(- 2, 4)$ .
A line passing through the points $(a, 2 a)$ and $(- 2, 3)$ is perpendicular to the line $4 x + 3 y + 5 = 0$ ; find the value of $a$ .
Show that the lines $y = 7 x + 2$ and $2 y - 14 x + 1 = 0$ are parallel.
Prove that the line $k^{2} x + k y + 1 = 0$ is perpendicular to the line $x - k y = 1$ for all real values of $k (k \neq 0)$ .
For what value of $k$ is the line $x - y + 2 + 𝑘 (2 x + 3 y) = 0$ parallel to the line $3 x + y = 0$ .
Prove that the lines $2 x - 3 y + 1 = 0, x + y = 3, 2 x - 3 y = 2$ and $x = 4 - y$ form a parallelogram.
Find the value of $θ$ between $0$ and $π$ if $x \cos θ + y \sin θ = 2$ is perpendicular to the line $x - y = 3$ .
If the line $x - 3 y + 5 + k (x + y - 3) = 0$ , where 𝑘 is arbitrary, is perpendicular to the line $x + y = 1$ , then find 𝑘 and the equation of the first line.
Prove that the median of an equilateral triangle is perpendicular to the corresponding side.
Prove that the diagonals of a rhombus are at right angles.
Find the equation of a line through $(3, 4)$ and parallel to the line $y = 3 x + 5$ .
Find the equation of the straight line through $(2, 3)$ and perpendicular to the line $4 x - 3 y = 10$ .
Find the equation of the straight line which has $y$ intercept equal to $\frac{4}{3}$ and is perpendicular to the line $3 x - 4 y + 11 = 0$ .
Find the equation of the perpendicular bisector of the line segment joining the points $(1, 2)$ and $(2, 3)$ .
The line $x + y = a$ meets the axes of $x$ and $y$ at $A$ and $B$ respectively. A $△ A M N$ is inscribed in the $△ O A B$ ( $O$ being the origin) with right angle at $N$ . $M, N$ respectively lie on $O B$ and $A B$ . If the area of the triangle $A M N$ is $\frac{3}{8}$ th of the area of the $△ O A M$ , then find $A N : N B$ .
Find the slope of the lines which make an angle of $45^{\circ}$ with the line $3 x - y + 5 = 0$ .
Find the equation of the lines through the point $(3, 2)$ which makes an angle of $45^{\circ}$ with the line $x - 2 y = 3$ .
A vertex of an equilateral triangle is $(2, 3)$ and the equation of the opposite side is $x + y = 2$ . Find the equation of the other sides of the triangle.
A line $4 x + y = 1$ through the point $A (2, - 7)$ meets the line $B C$ whose equation is $3 x - 4 y + 1 = 0$ at the point $B$ . Find the equation of the line $A C$ , so that $A B = A C$ .
Find the equation of straight lines passing through $(- 2, - 7)$ and having an intercept of length $3$ between the straight lines $4 x + 3 y = 12$ and $4 x + 3 y = 3$ .
Find the equation of the straight line parallel to $x + 2 y = 3$ and passing through the point $(3, 4)$ .
Find the equation of the straight line which passes through the point $(4, 3)$ and is parallel to the line $3 x + 4 y = 12$ .
Find the equation of the straight line parallel to $3 x - 4 y + 6 = 0$ and passing through the middle point of the line segment made by $(2, 3)$ and $(4, - 1)$ .
Find the equation of the straight line passing through the point $(2, 1)$ and parallel to the line joining the points $(2, 3)$ and $(3, - 1)$ .
Find the equation of the straight line passing through the point $(α, β)$ and parallel to the line $l x + m y + n = 0$ .
Find the equation of the straight line passing through the point $(2, 5)$ and perpendicular to the line $2 x + 5 y = 31$ .
Find the equation to the straight line which passes through the point $(x^{'}, y^{'})$ and is perpendicular to the line $y y^{'} = 2 a (x + x^{'})$ .
Find the angle between the straight lines $(m^{2} - m n) y = (m n + n^{2}) x + n^{3}$ and $(m n + m^{2}) y = (m n - n^{2}) + m^{2}$ .
Prove that the equation of the straight line which passes through the point $(a \cos^{3} θ, a \sin^{3} θ)$ and is perpendicular to the line $x \sec θ + y \csc θ = a$ is $x \cos θ - y \sin θ = a \cos 2 θ$ .
Find the equation of the straight line through $(a \cos θ, b \sin θ)$ perpendicular to the line $\frac{x}{a} \cos θ + \frac{y}{b} \sin θ = 1$ .
Two consecutive sides of a parallelogram are $4 x + 5 y = 0$ and $7 x + 2 y = 0$ . If the equation of one of the diagonal is $11 x + 7 y = 9$ , find the equation of the other diagonal.
Show that the area of the triangle whose sides are $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $x = 0$ is $\frac{1}{2} . \frac{(c_{2} - c_{1})^{2}}{m_{1} . m - 2}$ .
Show that the area of the triangle formed by the lines $y = m_{r} x + c_{r}, r = 1, 2, 3$ is $\frac{(c_{1} - c_{2})^{2}}{2 (m_{1} - m_{2})} + \frac{(c_{2} - c_{3})^{2}}{2 (m_{2} - m_{3})} + \frac{(c_{3} - c_{1})^{2}}{2 (m_{3} - m_{1})}$ .
Show that the area of the triangle whose sides are $a_{r} x + b_{r} y + c_{r} = 0, r = 1, 2, 3$ is $\frac{Δ^{2}}{2 | C_{1} C_{2} C_{3} |}$ , where $C_{1}, C_{2}$ and $C_{3}$ are the cofactors of $c_{1}, c_{2}$ and $c_{3}$ respectively in the determinant $Δ = - | \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} |$ .
Show that the lines $4 x + y - 9 = 0, x - 2 y + 3 = 0, 5 x - y - 6 = 0$ make equal intercepts on any line of gradient $2$ .
Find the coordinates of the foot of the perpendicular drawn from point $(2, 3)$ to the line $y = 3 x + 4$ .
Find the image of the point $(- 8, 12)$ with respect to the line mirror $4 x + 7 y + 13 = 0$ .
If the image of the point $(x_{1}, y_{1})$ with respect to the mirror $a x + b y + c = 0$ be $(x_{2}, y_{2})$ , show that $\frac{x_{2} - x_{1}}{a} = \frac{y_{2} - y_{1}}{b} = - \frac{2 (a x_{1} + b y_{1} + c_{1})}{a^{2} + b^{2}}$ .
A ray of light is sent along the lines $x - 2 y - 3 = 0$ . Upon reaching the line $3 x - 2 y - 5 = 0$ , the ray is reflected from it. Find the equation of the line containing the reflected ray.
A man starts from the point $P (- 3, 4)$ and will reach the point $Q (0, 1)$ touching the line $2 x + y = 7$ at $R$ . Find $R$ on the line sho that he will travel the shortest distance.
A ray of light is sent along the line $2 x - 3 y = 5$ . After refracting across the line $x + y = 1$ it enters the opposite side after turning by $15^{\circ}$ away from the line $x + y = 1$ . Find the equation of the line along which the refracted ray travels.
Find the equation of the straight line which passes through the point $(2, - 2)$ and the point of intersection of the lines $5 x - y = 9$ and $x + 6 y = 8$ .
Find the equation of the straight line which passes through the point of intersection of the lines $x - y - 1 = 0$ and $2 x - 3 y + 1 = 0$ and is parallel to the line $3 x + 4 y = 14$ .
Find the equation for the straight line which passes through the point of inter- section of the lines $3 x - 4 y - 7 = 0$ and $12 x - 5 y - 13 = 0$ and is perpendicular to the line $2 x - 3 y + 5 = 0$ .
Find the equation of the straight lines passing through the point of intersection of the lines $x + 3 y + 4 = 0$ and $3 x + y + 4 = 0$ and equally inclined to the axes.
The equation of two sides of a triangle are $3 c - 2 y + 6 = 0$ and $4 x + 5 y = 20$ and the orthocenter is $(1, 1)$ . Find the equation of the third side.
Show that the diagonal of the parallelogram whose sides are $u = p, u = q, v = r, v = s$ where $u = a x + b y + c$ and $v = a^{'} x + b^{'} y + c^{'}$ and which passes through the points of intersection of lines $u = p, v = 3$ and $u = q, v = s$ is given by $| \begin{matrix} u & v & 1 \\ p & r & 1 \\ q & s & 1 \end{matrix} | = 0$ .
Show that the straight lines $x (a + 2 b) + y (a + 3 b) = a + b$ pass through a fixed point for different values of $a$ and $b$ .
If $l x + m y + n = 0$ , where $l, m, n$ are variables is the equation of a variable line and $l, m, n$ are connected by the relation $a l + b m + c n = 0$ , where $a, b, c$ are constants, show that the line passes through a fixed point.
A variable line cuts $n$ given concurrent straight lines at $A_{1}, A_{2}, A_{3}, \dots, A_{n}$ , such that $\sum_{i = 1}^{n} \frac{1}{O A_{i}}$ is a constant. Show that it always passes through a fixed point. $O$ is the point of intersection of the lines.
Prove that the straight lines $4 x + 7 y = 9, 5 x - 8 y + 15 = 0$ and $9 x - y + 6 = 0$ are concurrent.
Prove analytically that medians of a triangle are concurrent.
Show that the lines $(p + q) 𝑥 + (p + q) y - (p - q) = 0, (p - q) x - (p - q) y - (p + q) = 0, p x + q y - p = 0$ and $q x + p y + q = 0$ are concurrent.
If the lines $p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1$ and 𝑝3𝑥 + 𝑞3𝑦 = 1 be concurrent, show that the points $(p_{1}, q_{1}), (p_{2}, q_{2})$ and $(p_{3}, q_{3})$ are collinear.
For what value of $m$ , the line $m x + 2 y + 5 = 0$ will pass through the point of intersection of the lines $x - 4 y = 3$ and $x + 2 y = 9$ ?
Find the point of intersection of the lines $y t_{1} = x + a t_{1}^{2}$ and $y t_{2} = x + a t_{2}^{2}$ .
If the straight line $\frac{x}{a} + \frac{y}{b} = 1$ passes through the point of intersection of the lines $x + y = 3$ and $2 x - 3 y = 1$ and is parallel to the line $y = x - 6$ , find $a, b$ .
Find the vertice and area of the triangle whose sides are $x = y, y = 2 x$ and $y = 3 x + 4$ .
Find the area of the triangle which is formed by the lines $3 x - 4 y + 4 a = 0, 2 x - 3 y + 4 a = 0$ and $5 x - y + a = 0$ .
Show that the area of the triangle formed by the three straight lines $y = m_{1} x, y = m_{2} x$ and $y = c$ is equal to $\frac{1}{4} c^{2} \sqrt{11} (\sqrt{3} + 1)$ , where $m_{1}, m_{2}$ are the roots of the equation $x^{2} + (\sqrt{3} + 2) + \sqrt{3} - 1 = 0$ .
Find the coordinates of the foot of the perpendicular drawn from the point $P (- 8, 12)$ on the line $4 x + 7 y + 13 = 0$ .
Find the projection of the point $(1, 0)$ on the line joining the points $P (- 1, 2)$ and $Q (5, 4)$ .
If perpendiculars are drawn from origin to the straight lines $x + 3 y = 3$ and $2 x + 3 y = 5$ , then find the equation of the line joining the foot of these perpendiculars.
If $(h, r)$ is the foots of the perpendiculars from $(x_{1}, y_{1})$ to $l x + m y + n = 0$ prove that $\frac{x_{1} - h}{l} = \frac{y_{1} - r}{m} = \frac{l x_{1} + m y_{1} + n}{l^{2} + m^{2}}$ .
find the image of the point $(- 8, 12)$ with respect to a line mirror $4 x + 7 y + 13 = 0$ .
If the image of the point $(2, 1)$ with respect to a line mirror be $(5, 2)$ , find the equation of the mirror.
Find the equation of the straight line which passes through the point $(1, 1)$ and the point of intersection of the lines $3 x + 2 y = 0$ and $x - 2 y = 0$ .
Find the equation of the straight line which passes through the point $(2, - 2)$ and the point of the intersection of the lines $5 x - y = 9$ and $x + 6 y = 8$ .
Find the equation of the straight line passing through the point of intersection of the lines $2 x + y - 1 = 0$ and $x + 3 y - 2 = 0$ and making with the coordinate axes a triangle of area $\frac{3}{8}$ .
The sides $A B$ and $A D$ of a parallelogram $A B C D$ are $2 x - y + 1 = 0$ and $x + 3 y - 10 = 0$ respectively and $C$ is the point $(- 1, - 2)$ . Find the equation of the diagonals $A C$ and $B D$ .
Prove that the lines $2 x - y - 5 = 0, 3 x - y - 6 = 0$ and $4 x - y - 7 = 0$ are concurrent.
Find the value of 𝑚 for which the two lines $m x + (2 m + 3) y + m + 6 = 0$ and $(2 m + 1) x + (m - 1) y + m - 9 = 0$ intersect at a point on 𝑦-axis.
Find the value of $m$ so that lines $y = x + 1, 2 x + y = 16$ and $y = m x - 4$ may be concurrent.
If the three lines $a x + a^{2} y + 1 = 0, b x + b^{2} y + 1 = 0$ and $c s + c^{2} y + 1 = 0$ are concurrent, show that at least two of the three constants $a, b, c$ are equal.
Find the condition that the lines $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $y = m_{3} x + c_{3}$ may be concurrent.
Show that the straight lines $(b + c) x + a y + 1 = 0, (c + a) x + b y + 1 = 0$ and $(a + b) x + c y + 1 = 0$ are concurrent.
Prove analytically that the right bisectors of the sides of a triangle are concurrent.
Prove that perpendiculars drawn from the vertices to the opposite sides are concurrent.
Prove that the family of lines represented by $x (1 + λ) + y (2 - λ) + 5 = 0, λ$ being arbitrary, pass through a fixed point. Also find the fixed point.
Prove that the line $x (a + 2 b) + y (a - 3 b) = a - b$ passes through a fixed point for different values of $a$ and $b$ .
Find the centroid and incenter of the triangle whose sides are $3 x - 4 y = 0, 5 x + 12 y = 0$ and $y - 15 = 0$ .
Find the coordinate of the orthocenter of the triangle whose vertices are $(0, 0), (2, - 1)$ and $(- 1, 3)$ .
Find the coordinate of the orthocenter of the triangle whose sides are $3 x - 2 y = 6, 3 x + 4 y + 12 = 0$ and $3 x - 8 y + 12 = 0$ .
Two vertices of a triangle are $(3, - 1)$ and $(- 2, 3)$ and its orthocenter is origin, find the coordinate of its third vertex.
A triangle has the lines $y = m_{1} x$ and $y = m_{2} x$ for two of its sides, where $m_{1}$ and $m_{2}$ are the roots of the equation $b x^{2} + 2 h x + a = 0$ . If $H (a, b)$ is the orthocenter of the triangle, show that the equation of the third side is $(a + b) (a x + b y) = a b (a + b - 2 h)$ .
A triangle is formed by the straight lines $a x + b y + c = 0, l x + m y + n = 0$ and $p x + q y + r = 0$ . Show that that straight line $\frac{p x + q y + r}{a p + b q} = \frac{l x + m y + n}{a l + n b}$ passes through the orthocenter of the triangle.
The three sides of a triangle are $L_{r} = x \cos θ_{r} + y \sin θ_{r} - p_{r} = 0, r = 1, 2, 3$ . Show that the orthocenter of the triangle is given by $L_{1} \cos (θ_{2} - θ_{3}) = L_{2} \cos (θ_{3} - θ_{1}) = L_{3} \cos (θ_{1} - θ_{2})$ .
Find the centroid and incenter of the triangle whose sides have the equations $3 x - 4 y = 0, 12 y + 5 x = 0$ and $y - 15 = 0$ .
The coordinates of the vertices $A, B$ and $C$ of the $△ A B C$ taken in anti-clockwise order are respectively $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ . Prove that the $∠ a$ is acute or obtuse according as $(x_{1} - x_{2}) (x_{1} - x_{3}) + (y_{1} - y_{2}) (y_{1} - y_{3}) > 0$ or $< 0$ . Also find the condition for the triangle to be right-angled at $y$ .
Show that the four lines $4 x - 3 y = 5, x - 2 y = 10, 7 x + y = 40$ and $x + 3 y + 10 = 0$ form the sides of a cyclic quadrilateral.
Find the condition for the quadrilateral to be cyclic whose sides are $a_{r} x + b_{r} y + c_{r} = 0; r = 1, 2, 3, 4$ taken in order.
Show that the lines $2 x + 3 y + 19 = 0$ and $9 x + 6 y - 17 = 0$ cut the coordinate axes in concyclic points.
Find the equation of the sides of a triangle having $B (- 4, - 5)$ as a vertex, $5 x + 3 y - 4 = 0$ and $3 x + 8 y + 13 = 0$ as the equation of two of altitudes not passing through $B$ .
The straight line $L$ is perpendicular to the line $5 x - y = 1$ . The area of the triangle formed by $L$ and the coordinate axes is $5$ . Find the equation of the line.
The line $2 x + 3 y = 12$ meets the $x$ -axis at $A$ and $y$ -axis at $B$ . The line through $(5, 5)$ perpendicular to $𝐴 𝐵$ meets the $x y$ axes and $A B$ at $C, D, E$ respectively. If $O$ is origin of axes then find the area of $O C E B$ .
A square has its center at origin and one vertex at $(1, 2)$ . Find the equation of its sides.
$A B C$ is an equilateral triangle. $A D$ is its altitude through $A$ . If $A = (1, 2)$ and $D = (- 2, 6)$ , find the equations of the sides of the triangle.
The equation of one side of an equilateral triangle is $x - y = 0$ and one vertex is $(2 + \sqrt{3}, 5)$ . Prove that second side is $y + (2 - \sqrt{3}) x = 6$ , and find the equation of the third side.
A diagonal of a square lies along the line $8 x - 15 y = 0$ and one vertex of the square is $(1, 2)$ . Find the equations of the lines of the square passing through this vertex.
Find the equation of the lines which pass through $(4, 5)$ andd make equal angles with the lines $5 y = 12 x + 6$ and $3 x = 4 y + 7$ .
Two equal sides of an isosceles triangle have the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third sides passes through the point $(1, - 10)$ . Determine the equation of the third side.
Prove that area of the triangle formed by the three straight lines $x \cos α + y \sin α - p_{1} = 0, x \cos β + y \sin β - p_{2} = 0$ and $x \cos γ + y \sin γ - p_{3} = 0$ is $\frac{1}{2} . \frac{{[p_{1} \sin (γ - β) + + p_{2} \sin (α - γ) + p_{3} \sin (β - α)]}^{2}}{\sin (γ - β) \sin (α - γ) \sin (β - α)}$ .
Find the area of a triangle formed by the 𝑦-axis, the straight line $L$ passing through the points $(1, 1)$ and $(2, 0)$ and the straight line perpendicular to the line $L$ and passing through $(\frac{1}{2}, 0)$ .
Find the coordinates of the feet of the perpendicular from the point $(9, 3)$ to the sides of the triangle whose vertices are at the points $(0, 0), (8, 0), (4, 8)$ . Prove that the points so determined lies on a straight line and find its equation.
Obtain the coordinates $(α, β)$ of the foot of the perpendicular from the origin to $\frac{x}{a} + \frac{y}{b} = 1$ and show that $(α^{2} + β^{2}) (α + β) = (a + b) α β$ .
Find the equation of the diagonal through the origin of the quadrilateral formed by $x = 0, y = 0, x + y = 1, 6 x + y = 3$ .
The altitudes of a $△ A B C$ are respectively $A D, B E, C F$ . If the points $A, D, E, F$ have the coordinates $(- 4, 5), (\frac{16}{5}, - \frac{23}{5}), (4, 1), (- 1, - 4)$ , find the coordinates of other vertices of the triangle.
Prove that the lines $y = m_{r} x + c_{r}; r = 1, 2, 3$ cut off equal intercepts on the transversal $x + y = 1 i f 1 + m_{1}, 1 + m_{2}, 1 + m_{3}$ are in H.P.
A line is such that its segment between the lines $5 x - y + 4 = 0 a n d 3 x + 4 y - 4 = 0$ is bisected at the point $(1, 5)$ . Obtain its equation.
If the lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ cut the coordinate axes in cyclic points, prove that $| a_{1} a_{2} | = | b_{1} b_{2} |$ .
A rectangle 𝐴𝐵𝐶𝐷 is inscribed in a circle with a diameter lying along the line $3 y = x + 10$ . If $A$ and $B$ are the points $(- 6, 7)$ and $(4, 7)$ respectively, then find the area of the rectangle.
From the point $(2, 5)$ rays of light are sent at $45^{\circ}$ with the line $2 x + y = 1$ . Find the equation of the lines of the reflected rays if the rays reflect from $x + 2 y = 1$ .
A ray of light is sent along the straight line $y = \frac{2 x}{3} - 4$ . On reaching the $x$ -axis it is reflected. Find the point of incidence and the equation of the reflected ray.
From the point $M (- 2, 3)$ a ray of light is sent at an angle $α$ to the positive direction of $x$ -axis. Upon reaching the $x$ -axis the ray is reflected from it. Find the equation of the reflected ray if $\tan α = 3$ .
A light beam emanating from the point $(3, 10)$ reflects from the straight line $2 x + y - 6 = 0$ and then passes through the point $B (7, 2)$ . Find the equations of the incident and reflected beams.
The line $3 x + 2 y = 24$ meets 𝑦-axis at $A$ and 𝑥-axis at $B$ . The perpendicular bisector of 𝐴𝐵 meets the line through $(0, - 1)$ parallel to $x$ -axis at $C$ . Find the area of the $△ A B C$ .
Find the condition that the real line $a x + b y + c = 0, b x + c y + a = 0$ and $c x + a y + b = 0$ are concurrent.
Find the condition that the lines $(b - c) x + (c - a) y + a - b = 0, (c - a) x + (a - b) y + b - c = 0$ and $(a - b) x + (b - c) 𝑥 + c - a = 0$ are concurrent.
Prove that the determinant $| \begin{matrix} x_{2} - x_{3} & y_{2} - y_{3} & x_{1} (x_{2} - x_{3}) + y_{1} (y_{2} - y_{3}) \\ x_{3} - x_{1} & y 32 - y_{1} & x_{2} (x_{3} - x_{1}) + y_{2} (y_{3} - y_{1}) \\ x_{1} - x_{2} & y_{1} - y_{2} & x_{3} (x_{1} - x_{2}) + y_{3} (y_{1} - y_{2}) \end{matrix} | = 0$ . What geometrical property does it imply for a triangle whose vertices are $(x_{r}, y_{r}); r = 1, 2, 3$ ?
Prove that all lines represented by the equation $(2 \cos θ + 3 \sin θ) 𝑥 + (3 \cos θ - 5 \sin θ) y - (5 \cos θ - 2 \sin θ) = 0$ pass through a fixed point for all values of $θ$ . Find the coordinates of that point and its reflection in the line $x + y = \sqrt{2}$ .
Prove that the orthocenter of the triangle formed by the three lines $y = m_{1} x + a l m_{1}, y = m_{2} x + a l m_{2}, y = m_{3} x + a l m_{3}$ is $[- a, (\frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} + \frac{1}{m_{1} m_{2} m_{3}})]$ .
If the coordinates of the point $P, Q, R$ satisfy the relation $x y = c^{2}$ , show that the orthocenter of $△ P Q R$ also satisfies the relation.
$A$ and $B$ are two fixed points $(3, 2)$ and $(5, 1)$ respectively. $△ A B P$ is equilateral and is situated on the side of $A B$ remote from the origin. Find the coordinates of $P$ and the orthocenter of the $△ A B P$ .
Vertices of a triangle are $A (x_{1}, x_{1} \tan α_{1}), B (x_{2}, x_{2} \tan α_{2})$ and $C (x_{3}, x_{3} \tan α_{3})$ . If the circumcenter coincide with the origin and the orthocenter $H$ is $(x, y)$ then prove that $\overset{―}{y} (\cos α_{1} + \cos α_{2} + \cos α_{3}) = \overset{―}{x} (\sin α_{1} + \sin α_{2} + \sin α_{3})$ , where $x_{1} \sec α_{1}, x_{2} \sec α_{2}, x_{3} \sec α_{3}$ have the same sign.
Find the area and the orthocenter of the triangle formed by the lines $x + l y = l^{2}, x + m y = m^{2}$ and $x + n y = n^{2}$ .
If the equation of the sides of a triangle are respectively $a_{1} x + b_{1} y = 1, a_{2} x + b_{2} y = 1$ and $a_{3} x + b_{3} y = 1$ and whose orthocenter is the origin, prove that $a_{1} a_{2} + b_{1} b_{2} = a_{2} a_{3} + b_{2} b_{3} = a_{3} a_{1} + b_{3} b_{1}$ .
Prove that the $△ D E F$ has the same centroid as $△ A B C$ , where $D, E, F$ are the middle points of the sides of the later triangle. Also prove that the orthocenter of the $△ D E F$ coincides with the circumcenter of the $△ A B C$ .
The circumcenter of a triangle with vertices $A (a, \tan α), V (b, \tan β)$ and $C (c, \tan γ)$ lies at the origin $α + β + γ = π$ , show that its orthocenter lies on the line $4 (\cos \frac{α}{2} + \cos \frac{β}{2} + \cos \frac{γ}{2}) x - 4 y \sin \frac{α}{2} \sin \frac{β}{2} \sin \frac{γ}{2} = y$ .
Show that the line $\frac{a_{x} + b_{1} y + c_{1}}{a_{1} a_{3} + b_{1} b_{3}} = \frac{a_{2} x + b_{2} y + c_{3}}{a_{2} a_{3} + b_{2} b_{3}}$ passes through the orthocenter of the triangle formed by the lines $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$ and $a_{3} x + b_{3} y + c_{3} = 0$ .
Find the position of the points $(1, 1)$ and $(2, - 1)$ with respect to the line $3 x + 4 y - 6 = 0$ .
Show that the four points $(0, 0), (- 1, 1), (- 7, - 4)$ and $(9, 6)$ are in the four different compartments made by the two straight lines $2 x - 3 y + 1 = 0$ and $3 x - 5 y + 2 = 0$ .
Find the position of the origin w.r.t. the triangle whose sides are $x + 1 = 0, 3 x - 4 y = 5, 5 x + 12 y = 27$ .
Show that the line segment joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is cut by the line $a x + b y + c = 0$ in the ratio $- \frac{a x_{1} + b y_{1} + c}{a x_{2} + b y_{2} + c}$ . Explain the minus sign.
A line $L$ intersects three sides $B C, C A$ and $A B$ of a $△ A B C$ in $P, Q$ and $R$ respectively. Prove that $B P . C Q . A R + P C . Q A . B R = 0$ .
Derive the condition to be imposed on $β$ so that $(0, β)$ should lie on or inside the triangle having sides $y + 3 x + 2 = 0, 3 y - 2 x = 5$ and $x + 4 y = 14$ .
A rhombus has two consecutive vertice at $(2, 3)$ and $(- 2, 6)$ and two of the sides are parallel to $2 x + y = 1$ . Find the other vertices of the rhombus if $(0, 3)$ is an interior point of rhombus.
Examine wheher the points $(3, - 4)$ and $(2, 6)$ are on the same side or opposite sides of the line $3 x - 4 y = 8$ ?
Prove that the points $(2, - 1)$ and $(1, 1)$ are on the opposite sides of the straight line $3 x + 4 y - 6 = 0$ .
Find the position of the points $(3, 4)$ and $(- 1, 1)$ w.r.t. the line $6 x + y - 1 = 0$ .
Prove that the points of intersection of the line $x - y = 2$ with the parallel lines $2 x + y = 7$ and $2 x + y = 16$ are on the opposite sides of the line $x + y = 5$ .
Find the distance of the point $(4, 5)$ from the straight line $3 x - 5 y + 7 = 0$ .
Find the distance of the point $(1, 2)$ from the straight line with slope $5$ and passing through the point of intersection of $x + 2 y = 5$ and $x - 3 y = 7$ .
The equation of the base of an equilateral triangle is $x + y = 2$ and the vertex is $(2, - 1)$ . Find the length of the side of the triangle.
If $a$ and $b$ are the intercepts of a straight line on the $x$ and $y$ axes respectively and $p$ be its perpendicular distance from the origin, prove that $\frac{1}{p^{2}} = \frac{1}{a^{2}} + \frac{1}{b^{2}}$ .
If $p$ and $p^{'}$ be the lengths of perpendiculars from origin to the lines $x \sec θ - y \csc θ = a$ and $x \cos θ - y \sin θ = a \cos 2 θ$ respectively, show that $4 p^{2} + p^{' 2} = a^{2}$ .
Find the equation of straight line which cuts off intercepts on $x$ -axis twice that on $y$ -axis and is at a unit distance from the origin.
Find the distance between the parallel lines $a x + b y + c = 0$ and $a x + b y + d = 0$ .
Prove that the length of the perpendiculars from points $(m^{2}, 2 m), (m n, m + n)$ and $(n^{2}, 2 n)$ to the line $x \cos θ + y \sin θ + \frac{\sin^{2} θ}{\cos θ}$ form a G.P.
A straight road passes through two towns, one $6$ km east and the other $2 \frac{1}{2}$ km north from a tower. Where should a rest house be constructed by the side of the road so that it may be nearest to the tower.
A straight line is such that the algebraic sum of the perpendiculars upon it from any number of fixed points is zero. Show that the line always passes through a fixed point.
The coordinates of the extremeties $A$ and $B$ of a rod are $(1, 2)$ and $(3, 4)$ respectively. $S (0, 0)$ is a point source of light. The rod $A b$ is parallel to the wall and is at equal distance from $S$ and the wall. If $C D$ is the shadow of $A B$ on the wall, find the coordinates of $C$ and D and the lnegth $C D$ . $S, A b$ and $C D$ are planar.
Prove that the diagonals of the parallelogram formed by the lines $\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1, \frac{x}{a} + \frac{y}{b} = 2$ and $\frac{x}{b} + \frac{y}{a} = 2$ are at right angles.
Find the area of the parallelogram whose sides are $y = m x + a, y = m x + b, y = n x + c$ and $y = n x + d$ .
Find the distance of the point of intersection of the lines $2 x + 3 y = 21$ and $3 x - 4 y + 11 = 0$ from the line $8 x + 6 y + 5 = 0$ .
Find the length of the perpendicular drawn from the origin upon line joining the points $(a, b)$ and $(b, a)$ .
Find the length of the perpendicular from the point $(4, - 7)$ to the line joining the origin and point of intersection of the lines $2 x - 3 y + 14 = 0$ and $5 x + 4 y - 7 = 0$ .
Find the equation of two straight lines which are parallel to $x + 7 y + 2 = 0$ , and at unit distance from the point $(1, - 1)$ .
Find the equations of the two straight lines parallel to $3 x - 4 y = 5$ at a unit distance from it.
Find the equation of two lines through $(0, a)$ which are at a distance 𝑎 from the point $(2 a, 2 a)$ .
Find the equation of the line through the point of intersection of the lines $x - 3 y + 1 = 0$ and $2 x + 5 y - 9 = 0$ and whose distance from origin is $\sqrt{5}$ .
Find the equation of the straight line passing the point of intersection of the lines $x - y + 1 = 0$ and $2 x - 3 y + 5 = 0$ and at a distance $\frac{7}{4}$ from the point $(3, 2)$ .
If the length of the perpendicular from the point $(1, 1)$ to the line $a x - b y + c = 0$ be $1$ , show that $\frac{1}{c} + \frac{1}{b} - \frac{1}{a} = \frac{c}{2 a b}$ .
Show that the product of the perpendiculars on the line $\frac{x}{a} \cos θ + \frac{y}{b} \sin θ = 1$ from the points $(\pm \sqrt{a^{2} - b^{2}}, 0)$ is $b^{2}$ .
Prove that the perpendicular distance between the lines $4 x + 3 y = 11$ is $8 x + 6 y = 15$ is $\frac{7}{10}$ .
Prove that the lines $2 x + 3 y = 19$ and $2 x + 3 y + 7 = 0$ are equidistant from the line $2 x + 3 y = 6$ .
Find the distance between the lines $y = m x + c$ and $y = m x + c_{1}$ .
The equation of two sides of a square whose area is $25$ square units are $3 x - 4 y = 0$ and $4 𝑥 + 3 𝑦 = 0$ . Find the equation of the other two sides of the square.
how that the parallelogram formed by $a x + b y + c = 0, a_{1} x + b_{1} y + c = 0, a x + b y + c_{1} = 0$ and $a - 1 x + b_{1} y + c_{1} = 0$ will be a rhombus if $a^{2} + b^{2} = a_{1}^{2} + b_{1}^{2}$ .
For the straight lines 4 x + 3 y − 6 = 0 and 5 x + 12 y + 9 = 0 , find the equation of the
1. bisector of the obtuse angle between them,
2. bisector of the acute angle between them, and
3. bisector of the angle which contains the origin.
Prove that the length of the perpendiculars drawn from any point of the line $7 x - 9 y + 10 = 0$ to the lines $3 x + 4 y - 5 = 0$ and $12 x + 5 y = 7$ are equal.
Prove that the internal bisectors of the angle of a triangle meet in a given point.
Find the coordinates of the incenter of the triangle whose sides are $x + 1 = 0, 3 x - 4 y - 5 = 0$ and $5 x + 12 y - 27 = 0$ .
Two opposite sides of a rhombus are $x + y = 1$ and $x + y = 5$ . If one vertex is $(2, - 1)$ and the angle at that vertex be $45^{\circ}$ , find the vertex opposite to given vertex.
Two sides of a rhombus 𝐴𝐵𝐶𝐷 are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ . If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on $y$ -axis, find the possible coordinates of $A$ .
Find the equations of the bisectors of the angle between the lines $x - 2 y + 3 = 0$ and $4 x + 2 y - 5 = 0$ .
Prove that the line $6 x + 66 y - 7 = 0$ is a bisector of the angle between the lines $15 x - 18 y - 1 = 0$ and $12 x + 10 y - 3 = 0$ .
Show that each point on the line $2 x + 11 y = 5$ is at equal distance from the lines $24 x + 7 y = 20$ and $4 x - 3 y = 2$ .
Find the locus of the point equidistant from the lines $6 x + 8 y - 10 = 0$ and $4 x - 3 y = 7$ .
Find the equations of the bisectors of the angles between the lines $x + y - 3 = 0$ and $7 x - y + 5 = 0$ and state which of them bisects the acute angle between the lines.
Prove that the bisector of the acute angle between the lines $3 x + 4 y = 11$ and $12 x - 5 y = 2$ is $11 x + 3 y = 17$ .
Find the equation of the line which bisects the obtuse angle between the lines $x - 2 y + 4 = 0$ and $4 x - 3 y + 2 = 0$ .
Show that the four points $(1, 0), (2, 3), (1, - 4)$ and $(8, 1)$ lie in the four compartments made by the lines $x + y - 2 = 0$ and $x - y - 3 = 0$ .
Determine whether the origin lies inside or outside the triangle whose saides are given by the equation $7 x - 5 y - 11 = 0, 8 x + 3 y + 31 = 0$ and $x + 8 y - 19 = 0$ .
Sides of a square lie on the line $5 x - 12 y - 65 = 0$ and $5 x - 12 y + 26 = 0$ . Find the area of the square.
The equation of two sides of a square are $3 x + 4 y - 5 = 0$ and $3 x + 4 y - 15 = 0$ . The third side has a point $(6, 5)$ on it. Find the equations of this and the remaining side of the square.
The equation of one side of rectangle is $3 x - 4 y - 10 = 0$ and the coordinates of two of its vertices are $(2, - 1)$ and $(2, 4)$ . Find the area of rectangle and the equation of that diagonal of the rectangle which passes through the point $(2, 4)$ .
Prove that the lines $a x \pm b y \pm c = 0$ enclose a rhombus whose area is $\frac{2 x^{2}}{| a b |}$ .
If $p, q, r$ be the lengths of perpendiculars from the vertices $A, B, C$ respectively of the $△ A B C$ on any straight line, then prove that $a^{2} (p - q) (p - r) + b^{2} (q - r) (q - p) + c^{2} (r - p) (r - q) = 4 Δ^{2}$ , where $Δ$ is the area of the triangle and $a, b, c$ length of the sides opposite to angles $A, B, C$ respectively.
Prove that no line can be drawn through the point $(4, - 5)$ so that its distance from $(- 2, 3)$ will be equal to $12$ .
The vertices of $△ O B C$ are $O (0, 0), B (- 3, - 1)$ and $C (- 1, - 3)$ . Find the equation of the line parallel of $B C$ and intersecting with $O B$ and $O C$ , whose perpendiculars distance from $O$ is $\frac{1}{2}$ .
The point $(1, - 1)$ is the center of a square, one of whose sides lies on the line $x - 2 y + 12 = 0$ . Find the equation of the straight lines which contain the remaining sides of the square.
The equation of two sides of a parallelogram are $3 x - 2 y + 12 = 0$ and $x - 3 y + 11 = 0$ and the point of intersection of its diagomals is $(2, 2)$ . Find the equation of other two sides and its diagonals.
Given three parallel lines $3 x + 4 y + 2 = 0, 3 x + 4 y + 5 = 0$ and $3 x + 4 y - 5 = 0$ . Show that the first of them lies between the other two. Also find the ratio in which the line divides the distance between the other two.
The three lines $x 𝑥 + 2 y + 3 = 0, x + 2 y - 7 = 0, 2 x - y - 4 = 0$ form the three sides of two squares. Find the equation of the fourth side of each square.
Find the equation of the internal bisectors of the angled of the triangle whose sides are $3 x + 4 y = 6, 12 x - 5 y = 3$ and $4 x - 3 y + 12 = 0$ .
Find the incenter of the triangle whose sides are $3 x + 4 y - 12 = 0, 5 x + 12 y - 20 = 0$ and $24 y - 7 x - 22 = 0$ .
Show that the reflection of the line $p x + q y + r = 0$ in the line $x + y + 1 = 0$ is the line $q x + p y + (p + q - r) = 0$ , where $p \neq - q$ .
A man at the corssing of two roads $x - 2 y - 4 = 0$ and $2 x - y - 4 = 0$ starts walking along the bisector of the acute angle between the roads and after covering a distance of $2$ km reaches the bank of a straight river at right angle to its path. Find the equation of the bank and the coordinates of the point where the path meets the bank.
A rhombus has two of its sides parallel to the lines $y = 2 x + 3$ and $y = 7 x + 2$ . If the diagonals cut at $(1, 2)$ and one vertex is on the $y$ -axis find the possible coordinates of that vertex.
A straight line segment of length 𝑙 moves with its end on two mutually perpendicular lines. Find the locus of the poitn which divides the segment in the ratio $1 : 2$ .
Find the locus of the middle point of the potion of the line $x \cos α + y \sin α = p$ which is intercepted between the axes given that $p$ remains constant.
A variable straight line, drawn through the point of intersection of the straight line $\frac{x}{a} + \frac{y}{b} = 12$ and $\frac{x}{b} + \frac{y}{a} = 1$ , meets the axes at $A$ and $B$ . Show that the locus of the mid-point of $A B$ is the curve $2 x y (a + b) = a b (x + y)$ .
If the line $\frac{x}{a} + \frac{y}{b} = 1$ moves in such a way that $\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c^{2}}$ , where $c$ is constant, prove that the foot of perpendicular from the origin on the straight line described the circle $x^{2} + y^{2} = c^{2}$ .
A straight line passes through a fixed point $(h, k)$ ; find the locus of the foot of the perpendicular on it drawn from the origin.
$O X$ and $O Y$ are two straight lines at right angles to one another. On $O Y$ a fixed point $A$ is taken and on $O X$ any point $B$ . On $A b$ an equilateral triangle is described, its vertex $C$ being on the side $A B$ away from $O$ . Show that the locus of the triangle is a straight line.
A point $P$ is such that its perpendicular distance from the line $y - 2 x + 1 = 0$ is equal to its distance from the origin. Find the equation of the locus of the point $P$ . Prove that the line $y = 2 x$ meets the locus in two points $Q$ and $R$ such that the origin is the mid-point of $Q R$ .
A line drawn through the origin intersect the lines $2 x + y - 2 = 0$ and $x - 2 y + 2 = 0$ in $A$ and $B$ . Find the locus of the mid-point of segment $A B$ .
$O$ is a fixed point and $A P$ and $B Q$ are two fixed parallel straight lines. $B O A$ is perpendicular to both and $∠ P O Q$ is a right angle. Prove that the locus of the foot of the perpendicular from $O$ on $P Q$ is a circle, whose diameter is $A B$ .
Two fixed points $P$ and $Q$ are given. $R$ is a variable point on one side of the line $P Q$ such that $∠ R P Q - ∠ R Q P$ is a positive constant $2 α$ . Find the locus of point $R$ .
A variable straight line passes through the points of intersection of the lines $x + 2 y = 1$ and $2 x - y =$ 1 and meets the coordinate axes at $A$ and $B$ . Find the locus of middle point of $A B$ .
Let $L_{1} = 0$ and $L_{2} = 0$ be two fixed lines. A variable line is drawn through the origin to cut the two lines at $A$ and $B$ . $P$ is a point on the line $A B$ such that $\frac{m + n}{O P} = \frac{m}{O R} + \frac{n}{O S}$ . Show that the locus of $P$ is a straight line through the point of intersection of the given lines $(R, S, P)$ are on the same side of origin.
Given $n$ straight lines and a fixed point $O$ . Through $O$ a straight line is drawn meeting these lines at the points $R R_{1}, R_{2}, \dots, R - n$ and a point $R$ is taken on it such that $\frac{n}{O R} = \frac{1}{O R_{1}} + \frac{1}{O R_{2}} + \dots + \frac{1}{O R_{n}}$ . Show that the locus of $R$ is a straight line.
The base of a triangle passes through a fixed point $(f, g)$ and its sides are respectively bisected at right angles by the lines $y^{2} - 8 x y - 9 x^{2} = 0$ . Determine the locus of its vertex.
Having given the bases and the sum of the areas of a number of triangles is constant, which have a common vertex, show that the locus of this vertex is a straight line.
If $A (\cos t \sin t), B (\sin t, - \cos t), C (1, 2)$ are the vertices of a $△ A B C$ , find the locus of centroid if $t$ varies.
The position of a moving point in the $x y$ -plane given at a time $t$ is given by $u \cos α, u \sin α - \frac{1}{2} g t^{2}$ , where $u, α, g$ are constants. Find the locus of the moving points.
A straight line passing through the point $(1, 1)$ is terminated by the axes of coordinates. Show that the locus of the mid-point of the line has the equation $2 x y = x + y$ .
Find the locus of the middle point of the intercepts made by the axes on the lines drawn through the point $(α, β)$ .
A straight line moves such that the sum of its intercepts on the axes is $k$ . Find the locus of the middle point of the portion of the line intercepted between the axes.
A line $A P B$ of constant length meets the $x$ -axis at $A$ and $y$ -axis at $B$ . If $A P = b, P B = a$ and the line slides with its extremities on the coordinate axes, show that the equation of the locus of the point $P$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .
A variable line through the point $(\frac{6}{5}, \frac{6}{5})$ cuts the coordinate axes at points $A$ and $B$ . If the point $P$ divides $A B$ internally in the ratio $2 : 1$ , show that the equation of the locus of $P$ is $5 x y = 2 (2 x + y)$ .
A straight line moves in such a way that the length of the perpendicular upon it from the origin is always $p$ . Find the locus of the centroid of the triangle which is formed by the line and the axes.
Two fixed points $A$ and $B$ have coordinates $(x_{1}, y_{1})$ and $(_{2}, y_{2} x)$ . A point $P$ moves such that $A P$ is perpendicular to $B P$ . Show that the locus of $P$ is $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$ .
A point moves so that the square of its distance from the point $(3, - 2)$ is numerically equal to its distance from the line $5 x - 12 y = 13$ . Find the equation of its locus.
A point moves such that the sum of its distance from two fixed points $(a e, 0)$ and $(- a e, 0)$ is always $2 a$ . Prove that its locud is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} (1 - e^{2})} = 1$ .
Find the locus of the middle point of the intercept on the line $y = x + c$ made by the line $2 x + 3 y = 5$ and $2 x + 3 y = 8, c$ being a parameter.
If a line $A B$ of length $2 l$ moves with end $A$ always on $x$ -axis and the end $B$ always on the line $y = 6 x$ . Find the equation of the locus of the mid-point of $A B$ .
$P$ is the point $(- 1, 2)$ . A variable line through $P$ cuts the coordinate axes at $A a$ and $B$ respectively. $Q$ is a point on $A B$ such that $P A, P Q, P B$ are in H.P. Show that the locus of $Q$ is the line $y = 2 x$ . ( $A, B$ lie on the same side of $P$ ) Also show that in general locus of $Q$ is the rhombus whose sides are $y = 2 x, y = - 2 x + 4, y = - 2 x - 4$ and $y = 2 x + 8$ excluding the vertices.
If $O$ is the origin, $A$ is the point $(4, 4)$ , $B$ is any point on the plane, find the locus of the point of intersection of the perpendicular bisectors of $O B$ and $A B$ .
Two fixed points $A$ and $A$ are taken on the two axes such that $O A = a$ and $O B = b$ . Two variable points $C$ and $D$ are taken on the same axes respectively, find the locus of the point of intersection of $A D$ and $B C$ if $\frac{1}{O C} - \frac{1}{O D} = \frac{1}{O A} - \frac{1}{O B}$ .
$Q$ is any point on the line $x = a$ . If $A$ is the fixed point $(a, 0)$ and $Q R$ , the bisector of the angle $O Q A$ , meets the $x$ -axis in $R$ , find the locus of the foot of the perpendicular from $R$ to $O Q$ .
A right-angled $△ A B C$ having a right angle at $C$ , $C A = b$ and $C B = a$ moves such that the angular points $A$ and $B$ slide along $x$ -axis and $y$ -axis respectively. Find the locus of $C$ .
Show that the locus of a point which moves such that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distance from the other sides, is a circle.
A variable straight line is drawn through a given point $O$ to cut two fixed straight lines in $R$ and $S$ ; on it is taken a point $P$ such that $\frac{2}{O P} = \frac{1}{O R} + \frac{1}{O S}$ , show that the locus of $P$ is a third fixed straight line.
If $p, x_{1}, x_{2}, \dots, x_{i}, \dots$ and $q, y_{1}, y_{2}, \dots, y_{i}, \dots$ from two infinite arithmetic sequences with common difference $a$ and $b$ respectively then find the locus of the point $(α, b e t a)$ where $α = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ and $β = \sum_{i = 1}^{n} y_{i}$ .

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