Chapter 4. Pair of Straight Lines

Index
Prev	Table of Contents	Next

4.1. Homogeneous Equations
4.2. Pair of Straight Lines Through Origin
4.3. Angle Between the Lines Represented by Second Degree Equation
4.4. Bisectors of Angles Between the Pair of Straight Lines
4.5. Condition for General Equation of Second Degree to Represent a Pair of Straight Lines
4.6. Problems

Consider a pair of straight lines $a x + b y + c = 0$ and $a_{1} x + b_{1} y + c_{1} = 0$ , which we represent as $(a x + b y + c) (a_{1} x + b_{1} y + c_{1}) = 0$ .

Let $P (α, b e t a)$ on the line $a x + b y + c = 0 \Rightarrow a α + b β + c = 0$ and $(a α + b β + c)$ $(a_{1} α + b_{1} β + c_{1}) = 0$ .

Thus, a point which lies on either line wil satisfy $(a x + b y + c) (a_{1} x + b_{1} y + c_{1}) = 0$ . This equation represents a pair of straight lines.

4.1. Homogeneous Equations

Any equation in which combined powers of $x$ and $y$ is constant(say $n$ ) is called a homogeneous equation of degree $n$ . For example, $a x^{2} 2 + 2 h x y + b y^{2} = 0$ is a homogeneous equation of degree $2$ .

4.2. Pair of Straight Lines Through Origin

We will show that any homogeneous equation of second degree in $x$ and $y$ represents a pair of straight lines through the origin.

Consider the equation $a x^{2} + 2 h x y + b y^{2} = 0 \Rightarrow a + b (\frac{y}{x}) 62 + 2 h \frac{y}{x} = 0$ , which is a quadratic equation in $\frac{y}{x}$ .

Let the roots be $m_{1}$ and $m_{2}$ of the above equation. $\Rightarrow \frac{y}{x} = m_{1}, m_{2} \Rightarrow b (y - m_{1} x) (y - m_{2} x) = 0.$

Thus, the given homogeneous equation represents two straight lines through origin.

If the lines are represented by $a x^{2} + 2 h x y + b y^{2} = 0$ and $y - m_{1} x = 0$ and $y - m_{2} x = 0$ , then

m_{1} + m_{2} = - \frac{2 h}{b}

(4.1)

and $m_{1} m_{2} = \frac{a}{b}$ (4.2)

4.3. Angle Between the Lines Represented by Second Degree Equation

The tangent of the angle between the two lines is given by $\tan θ = \pm \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} = \pm \frac{\sqrt{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}}}{1 + m_{1} m_{2}} = \pm \frac{2 \sqrt{h^{2} - a b}}{a + b}$

For lines to be perpendicular $a + b = 0$ (4.3) and for them to be paralle $h^{2} = a b$ (4.4)

4.4. Bisectors of Angles Between the Pair of Straight Lines

Coninuing from previous sections the equation of bisectors of straight lines is given by $\frac{y - m_{1} x}{\sqrt{1 + m_{1}^{2}}} = \pm \frac{y - m_{2} x}{\sqrt{1 + m_{2}^{2}}}$

Combined equation is given by $(\frac{y - m_{1} x}{\sqrt{1 + m_{1}^{2}}} - \frac{y - m_{2} x}{\sqrt{1 + m_{2}^{2}}}) (\frac{y - m_{1} x}{\sqrt{1 + m_{1}^{2}}} + \frac{y - m_{2} x}{\sqrt{1 + m_{2}^{2}}}) = 0$

$\Rightarrow \frac{(y - m_{1} x)^{2}}{\sqrt{(1 + m_{1}^{2})}} - \frac{(y - m_{2} x)^{2}}{(1 + m_{2}^{2})} = (m_{1} + m_{2}) (x^{2} - y^{2}) + 2 (m_{1} m_{2} - 1) x y = 0$

$\Rightarrow - \frac{2 h}{b (x^{2} - y^{2})} + 2 (\frac{a}{b} - 1) x y = 0 \Rightarrow \frac{x^{2} - y^{2}}{a - b} = \frac{x y}{h}$ $\frac{x^{2} - y^{2}}{a - b} = \frac{x y}{b}$ (4.5)

4.5. Condition for General Equation of Second Degree to Represent a Pair of Straight Lines

General equation of the second degree is given by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ . This equation will result into two straight lines if it can be resolved into two linear factors.

Let the straight lines be $l_{1} x + m_{1} y + n_{1} = 0$ and $l_{2} x + m_{2} y + n_{2} = 0$ , which are represented by the above equation.

Then $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = (l_{1} x + m_{1} y + n_{1}) (l_{2} x + m_{2} y + n_{2})$

Comparing coefficients $l_{1} l_{2} = a, m_{1} m_{2} = b, n_{1} n_{2} = c, m_{1} n_{2} + m_{2} n_{2} = 2 f, n_{1} l_{2} + n_{2} l_{1} = 2 g, l_{1} m_{2} + l_{2} m_{1} = 2 h$

Multiplying last three we obtain, $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$ which is the required condition.

The above condition can be represented as $| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} | = 0$ (4.6)

The general equation can be treated as an equation in 𝑥 and then discriminat must be a perfect square for it to resolve into linear factors. We will obtain the same condition as above using this method as well.

4.6. Problems

Find the joint equation of the straight lines represented by $x = 4 y - 2$ and $x - 1 = y$ .
Prove that the equations to the straight lines passing through the origin and making an angle $α$ with the straight line $y + x = 0$ are given by $x^{2} + 2 x y \sec 2 α + y^{2} = 0$ .
A pair of perpendicular straight lines are drawn through the origin forming with the line $2 x + 3 y = 6$ an isosceles triangle at the origin. Find the equation of pair of straight lines and the area of the triangle.
ind the combined equation of the lines $2 x - y = 3$ and $y = 3 x + 4$ .
Find the joint equation of the lines through $(1, 2)$ and parallel to the lines $x - 2 y = 5$ and $x = 3 y - 4$ .
Find the combines equation of the lines bisecting the angles between $x$ and $y$ axes.
Prove that the equation $8 x^{2} + 8 x y + 2 y^{2} + 26 x + 13 y + 15 = 0$ represents two straight lines.
If the equation $6 x^{2} + 2 k x y + 12 y^{2} + 22 x + 31 y + 20 = 0$ represent a pair of straight lines, find the value of $k$ .
Show that the equation $10 x 2 - 11 x y - 6 y^{2} - 12 x - y + 2 = 0$ represents a pair of straight lines.
Does the equation $2 x^{2} - 15 x y - 17 y^{2} + 4 x + 23 y - 6 = 0$ represent a pair of straight lines?
For what value of 𝑚 does the equation $m x^{2} - 5 x y - 6 y^{2} + 14 x + 5 y + 4 = 0$ represent two straight lines? Prove that they are perpendicular to each other.
For what values of 𝑚 does the equation $x^{2} 2 + m x y - 2 y^{2} + 3 y - 1 = 0$ represent two straight lines?
If the equation $12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + m = 0$ represent two straight lines, find the value of $m$ .
For what value of $m$ does the equation $6 x^{2} + 5 x y - 4 y^{2} + 7 x + m y + 2 = 0$ represent two straight lines?
Find the angle between the pair of straight lines represented by the equation $4 x^{2} + 24 x y + 11 y^{2} = 0$ .
Find the angle between the pair of straight lines represented by the equation $(x^{2} + y^{2}) \sin^{2} α = (x \cos β - y \sin β)^{2}$ .
Find the angle between the pair of straight lines represented by the equation $x^{2} - 5 x y + 4 y^{2} + 3 x - 4 = 0$ .
Show that the two straight lines $x^{2} (\tan^{2} θ + \cos^{2} θ) - 2 x y \tan θ + y^{2} \sin^{2} θ = 0$ make with the axis of $x$ angles such that the difference of their tangents is $2$ .
Find the length of the straight line joining the foot of perpendicular from the point $(𝑝, 𝑞)$ on the pair of lines $a x^{2} + 2 h x y + b y^{2} = 0$ .
A point moves so that the distance between the feet of the perpendiculars drawn from it to the lines $a x^{2} + 2 h x y + b y^{2} = 0$ is a constant $k$ . Show that the equation of its locus is $(x^{2} + y^{2}) (h^{2} - a b) = k^{2} [(𝑎 - 𝑏)^{2} + 4 h^{2}]$ .
Find the angle between the pair of straight lines given by $x^{2} - 3 x y - y^{2} = 0$ .
Find the angle between the lines given by $x^{2} 2 + 2 x y \cot 2 α - y^{2} = 0$ .
Find the angle between the lines given by $x^{2} - 2 p x y + y^{2} = 0$ .
Show that the two straight lines given by $x^{2} - 2 x y \sec θ + y^{2} = 0$ make an angle $θ$ with one another.
Find the angle between the lines represented by $(x^{2} + y^{2}) \sin^{2} α = (x \cos α - y \sin α)^{2}$ .
Prove that the equation $6 x^{2} - 5 x y - 6 y^{2} + 14 x + 5 y + 4 = 0$ represents two straight lines, which are perpendicular to each other.
Prove that the equation $16 x^{2} + 24 x y + 9 y^{2} + 40 x + 30 y - 75 = 0$ represents two parallel straight lines.
Prove that the equation $x^{2} - 5 x y + 4 y^{2} + x + 2 y - 2 = 0$ represents two straightdlines. Also, find the angle between them.
If the equation $12 x^{2} + 7 x y - p y^{2} - 18 x + q y + 6 = 0$ represents two perpendicular straight lines, find the values of $p$ and $q$ .
Prove that the equation $2 x^{2} + 3 x y - 2 y^{2} = 0$ represents two lines through origin which are perpendicular to each other.
Find the separate equation of the lines represented by $2 x^{2} - x y - y^{2} + 9 x - 3 y + 10 = 0$ .
Prove that the equation $2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 4 = 0$ represents a pair of straight lines. Find the coordinates of their point of intersection and also the angle between them.
Show that the equation $8 x^{2} + 8 x y + 2 y^{2} + 26 x + 13 y + 15 = 0$ represents a pair of parallel straight lines. Also, find the perpendicular distance between them.
Find the combined equation of the straight lines passing through the point $(1, 1)$ and parallel to the lines represented by the equation $x^{2} 2 - 5 x y + 4 y^{2} + x + 2 y - 2 = 0$ and find the angle between them.
If the lines represented by $2 x^{2} - 5 x y + 2 y^{2} = 0$ be the two sides of a parallelogram and the line $5 x + 2 y = 1$ be one of its diagonals, find the equation of the other diagonal and area of the parallelogram.
The base of a triangle passes through a fixed point $(f, g)$ and its sides are bisected at right angle by a pair of straight lines $y^{2} - 8 x y - 9 x^{2} = 0$ . Determine the locus of its vertex.
Prove that the straight lines represented by $(y - m x) 2 = a^{2} (1 + m^{2})$ and $(y - n x)^{2} = a^{2} (1 + n^{2})$ form a rhombus.
If the equation $2 h x y + 2 g x + 2 f y + c = 0$ represents two straight lines, show that they form a rectangle of area $\frac{| f g |}{h^{2}}$ with the coordinate axes.
Find separate equations of lines represented by $x^{2} 2 - 5 x y + 4 y^{2} + x + 2 y - 2 = 0$ .
Prove that the lines represented by $x^{2} 2 - 6 x y + 3 y^{2} = 0$ are perpendicular to the lines represented by $3 x^{2} + 6 x y + y^{2} = 0$ .
Show that one of the lines given by $y^{2} + x y - 12 x^{2} = 0$ coincides with one of the lines given by $4 y^{2} - 13 x y + 3 x^{2} = 0$ and the other two lines are perpendicular to one another.
Prove that the lines represented by $x^{2} 2 - 7 x y + 12 y^{2} = 0$ are perpendicular to the lines represented by $12 x^{2} + 7 x y + y^{2} = 0$ .
The equations to a pair of opposite sides of a rectangle are $x^{2} - 7 x + 6 = 0$ and $y^{2} - 14 y + 40 = 0$ . Find the equation of its diagonals.
Show that the four lines given by the equation $3 x^{2} + 8 x y - 3 y^{2} = 0$ and $3 x^{2} + 8 x y - 3 y^{2} + 2 x - 4 y - 1 = 0$ form a square. Also find the equation of the diagonals of the square.
Prove that the equation $2 x^{2} - 5 x y - 3 y^{2} - 2 x + 6 y = 0$ represents two straight lines and find their point of intersection.
Find the condition that one of the lines given by $a x^{2} + 2 x y + b y^{2} = 0$ may be perpendicular to one of the lines given by $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} = 0$ .
If the slope of one of the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ be $λ$ times the other, prove that ${(\frac{1 + λ}{2 h})}^{2} = \frac{λ}{a b}$ .
If the slope of one of the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ be the square of the other, prove that $\frac{a + b}{h} + \frac{8 h^{2}}{a b} = 6$ .
Find the condition that the pair of straight lines $a x^{2} + 2 h x y + b y^{2} = 0$ and $a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} = 0$ have one line in common.
Find the equation of the bisectors of angles between the lines $3 x^{2} - 5 x y + 4 y^{2} = 0$ .
Show that the straight lines represented by $135 x^{2} - 136 x y + 33 y^{2} = 0$ are equally inclined to the line $x + 2 y = 7$ .
Prove that the lines $a^{2} x^{2} + 2 h (a + b) x y + b^{2} y^{2} = 0$ are equally inclined to the lines $a x^{2} + 2 h x y + b y^{2} = 0$ .
Show that the lines bisecting the angles between the bisectors of the angles made by lines $a x^{2} + 2 h x y + b y^{2} = 0$ are $(a - b) (x^{2} - y^{2}) + 4 h x y = 0$ .
If pairs of straight lines $x^{2} - 2 p x y - y^{2} = 0$ and $x^{2} 2 - 2 q x y - y^{2} = 0$ be such that each pair bisects the other pair, prove that $p q = - 1$ .
Prove that the bisectors of the angles between the lines represented by $a x^{2} + 2 h x y + b y^{2} = λ (x^{2} + y^{2})$ are always the same irrespective of $λ$ .
Prove that the bisectors of the angle between the lines $a x^{2} + a c x y + c y^{2} = 0$ and $l r (3 + \frac{1}{c}) x^{2} + x y + (3 + \frac{1}{a}) y^{2} = 0$ are always the same.
Prove that the lines $2 x^{2} + 6 x y + y^{2} = 0$ are equally inclined to the lines $4 x^{2} + 18 x y + y^{2} = 0$ .
If the lines represented by $x^{2} - 2 p x y - y^{2} = 0$ are rotated about the origin through an angle $θ$ , one in clockwise direction and other in anti-clockwise direction, find the equation of the bisectors of the angles between the lines in the new position.
If one of the lines $a x^{2} + 2 h x y + b y^{2} = 0$ be the bisector of the angle between the coordinate axes, prove that $(a + b)^{2} = 4 h^{2}$ .
Find the equation of pair of lines both of which pass through $(1, 2)$ and are parallel to the bisectors of the angles between the lines given by $x^{2} + x y - 2 y^{2} + 4 x - y + 3 = 0$ .
Show that the lines joining the origin to the points common to $x^{2} + h x y - y^{2} + g x + f y = 0$ and $f x - g y = λ$ are at right angles irrespective of value of $λ$ .
Prove that the angle between the lines joining the origin to the points of intersection of the straight line $y = 3 x + 2$ with the curve $x^{2} + 2 x y + 3 y^{2} + 4 x + 8 y - 11 = 0$ is $\tan^{- 1} \frac{2 \sqrt{2}}{3}$ .
Prove that the pair of lines joining the origin to the intersection of the curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ by the line $l x + m y + n = 0$ are conincident if $a^{2} l^{2} + b^{2} m^{2} = n^{2}$ .
Show that the straight lines joining the origin to the other two points of intersection of the curves whose equations are $a x 62 + 2 x y + 𝑏 y^{2} + 2 g x = 0$ and $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0$ will be at right angles if $g (a_{1} + b_{1}) = g_{1} (a + b)$ .
Find the equation of the straight lines joining the origin to the point of intersection of the line $3 x + 4 y = 5$ and the curve $2 x^{2} + 3 y^{2} = 5$ .
Prove that the lines joining the origin and the points of intersection of the line $3 x - 2 y = 1$ and the curve $3 x^{2} + 5 x y - 3 y^{2} + 2 x + 3 y = 0$ are perpendicular to each other.
Find the equation of the straight lines joining the origin to the points of intersection of the line $y = m x + c$ and the curve $x^{2} + y^{2} = a^{2}$ . Prove that they are perpendicular to one another if $2 c^{2} = a^{2} (1 + m^{2})$ .
Find the equation of the straight lines joining the origin to the points of intersection of the line $l x + m y + n = 0$ and $y^{2} = 4 a x$ . Also find the condition of their perpendicularity.
Find the value of $c$ so that the lines joining the origin to the common points of $(x - 3)^{2} + (y - 4)^{2} = c^{2}$ and $4 x + 3 y = 24$ are at right angles.
Find the value of $m$ , if the lines joining the origin and the points of intersection of $y = m x + 1$ and $x^{2} + y^{2} = 1$ are perpendicular to one another.
Prove that the straight lines joining the origin to the points of intersection of the straight lines $k x + h y = 2 h k$ with the curve $(x - h)^{2} + (y - k)^{2} = c^{2}$ are at right angles if $h^{2} + k^{2} = c^{2}$ .
Find the equation to the pair of lines through the origin and perpendicular to the pair of lines $a x^{2} + 2 h x y + b y^{2} = 0$ .
Find the condition that the slope of one of the lines represented by $a x^{2} 2 + 2 h x y + b y^{2} = 0$ should be $λ$ times the slope of another.
Find the product of the length of the perpendiculars drawn from $(x_{1}, y_{1})$ on the pair of straight lines $a x^{2} + 2 b x y + b y^{2} = 0$ .
Prove that the equation $(a + 2 h + b) x^{2} - 2 (a - b) x y + (a - 2 h + b) y^{2} = 0$ represents a pair of lines inclined at an angle of $45^{\circ}$ to one or other of the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ .
If the distance of a given point $(α, β)$ from each of the two straight lines through the origin is $d$ , show that $(α y - β x)^{2} = d^{2} (x^{2} + y^{2})$ .
If one of the lines given by $a x^{2} 2 + 2 h x y + b y^{2} = 0$ coincides with one of those given by $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} = 0$ and the other lines represented by them be perpendicular, prove that $\frac{b a_{1} b_{1}}{b_{1} - a_{1}} = \frac{h a b}{a - b} = \frac{1}{2} \sqrt{- a a_{1} b b_{1}}$ .
If the equation $a x^{2} 2 + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represented a pair of parallel lines, prove that $\frac{a}{h} = \frac{h}{b} = \frac{g}{f}$ . Also prove that the distance between these parallel lines is $2 \sqrt{\frac{g^{2} - c a}{a (a + b)}}$ .
If the equation $a x^{2} 2 + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents a pair of straight lines, prove that the square of the distance of their point of intersection from the origin is $\frac{c (a + b) - f^{2} - g^{2}}{a b - h^{2}}$ .
If the equation $a x^{2} 2 + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents a pair of straight lines equidistant from the origin, prove thta $f^{4} - g^{4} = c (b f^{2} - a g^{2})$ .
If the lines $a x^{2} + 2 h x y + b y^{2} = 0$ be two sides of a parallelogram and the line $l x + m y = 1$ be one of its diagonals, show that the equation of the other diagonal is $y (b l - h m) = 𝑥 (a m - h l)$ .
Show that the orthocenter of the triangle formed by the lines $a x^{2} + 2 h x y + b y^{2} = 0$ and $l x + m y = 1$ is given by $\frac{x}{l} = \frac{y}{m} = \frac{a + b}{a m^{2} - 2 h l m + b l^{2}}$ .
Find the area of the triangle formed by the lines $a x^{2} + 2 h x y + b y^{2} = 0$ and $l x + m y + n = 0$ .
Show that the straight lines $(A^{2} - 3 B^{2}) x^{2} + 8 A B x y + (B^{2} - 3 A^{2}) y^{2} = 0$ form with the line $A x + B y + C = 0$ an equilateral triangle whose area is $\frac{C^{2}}{\sqrt{3} (A^{2} + B^{2})}$ .
Prove that the lines $(l x + m y)^{2} - 3 (m x - l y)^{2} = 0$ and $l x + m y + n = 0$ form an equilateral triangle.
Show that the four straight lines given by $12 x^{2} + 7 x y - 12 y^{2} = 0$ and $12 x^{2} + 7 x y - 12 y^{2} - x + 7 y - 1 = 0$ lie along the sides of a square.
The lines represented by $x^{2} - 3 x y + 2 y^{2} = 0$ are shifted parallel to itself so that their point of intersection comes to $(1, 1)$ . Find the combined equation of the lines in new position.
The joint equation of the lines of rays of incidence and reflection is $2 x^{2} - x y - y^{2} = 0$ . Find the joint equation of two possible lines from which the ray has been reflected.
If the angle between the lines joining the origin to the points of intersection of the lines $x \cos α + y \sin α = 1$ and the circle $x^{2} + y^{2} = a^{2}$ be $90^{\circ}$ , find the possible values of $a$ .
If the pair of straight lines $a x^{2} + 2 h x y + b y^{2} = 0$ is rotated about the origin through $90^{\circ}$ , find the equation in the new position.
Find the image of the pair of lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ by the line mirror whose equation is $y = 0$ .
Prove that the sum of the squares of the perpendiculars drawn from the point $(x^{'}, y^{'})$ on the lines given by $a x^{2} + 2 h x y + b y^{2} = 0$ is
$\frac{[4 h^{2} (x^{' 2} + y^{' 2}) + 4 h (a + b) x^{'} y^{'} + 2 (a - b) (a x^{' 2} - b y^{' 2})]}{[{(a - b)}^{2} + 4 h^{2}]}$ .
If $(x, y)$ be the centroid of the triangle whose sides are the lines $a x^{2} + 2 h x y + b y^{2} = 0$ and $l x + m y + n = 0$ . Find the centroid.
A triangle has the line $a x^{2} + 2 h x y + b y^{2} = 0$ for two of its sides and the point $(l, m)$ for its orthocenter. Prove that the third side has the equation $(a + b) (l x + m y) = a m^{2} - 2 h l m + b l^{2}$ .
Prove that the lines $x^{2} + 4 x y + y^{2} = 0$ and $x - y = 4$ are the sides of an equilateral triangle. Find its area.
Find the internal angles of the triangle formed by the pair of straight lines $x^{2} + 4 x y + y^{2} = 0$ and straight line $x + y + 4 \sqrt{6} = 0$ .
Prove that the area of the triangle formed by the lines $a x^{2} + 2 h x y + b y^{2} = 0$ and $x \cos α + y \sin α = p$ is $\frac{p^{2} \sqrt{h^{2} - a b}}{b \cos^{2} α - 2 h \cos α \sin α + a \sin^{2} α}$ .
Find the area of the triangle formed by the lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ and the $x$ -axis.

Home