Chapter 1. Coordinates

Index
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1.1. Number Line

1.2. Coordinates

1.3. Cartesian Coordinate System

1.4. Distance Between Two Points

1.5. Section Formula

1.6. Area of a Trapezium

1.7. Area of a Triangle

1.7.1. Sign of Area of a Triangle
1.7.2. Condition of Collinearity of Three Points

1.8. Centroid of a Triangle

1.9. Incenter of a Triangle

1.10. Area of a Quadrilateral

1.11. Polar Coordinates

1.12. Polar Coordinates and Cartesian Coordinates

1.13. Problems

Coordinate geometry is study of geometric using algebra. It is also called Analytical Geometry. In geometry, we study about points, lines, triangles, quadrilaterals, circles, polygons etc. without the use of algebra but as said in coordinate geometry these geometrical figures are studied using variables and equations of algebra. In coordinate geometry we will come to know a term called Cartesian plane or cartesian coordinates, which are named after René Des Cartes who first published his work on coordinate geometry in 1637. Pierre de Fermat also independently discovered, but he did not publish his discovery.

1.1. Number Line

We take a straight line and any point $O$ on it. This point $O$ is taken such that it divides the line in two equal parts. We take the part right of this point as positive part and the part left of it as negative part of this line. We represent the number $1$ with the point $O$ . Let us take another point $A$ such that it represents $1$ on the number line. Now $O A$ represents a unit, and thus we can represent all natural numbers in terms of it. We can also represent all real numbers on this number line. Positive real numbers will lie on the right side and negative real numbers will lie on the left side of the mid- point, which is $O$ . Because all real numbers can be represented on this line we call it real line or in general number line.

Figure 1.1. Number line

As we know from our basic algebra that there are a infinite number of rational and irrational numbers between any two numbers so we will have infinite points between any two points on this number line which we know from geometry.

1.2. Coordinates

Consider any point $P$ in a $2$ -dimensional plane. To define the location of this point we need a reference. In any $n$ -dimensional plane no point can be given an absolute position rather we define location of points in relation to some other point in the same plane. Typically, we define axes of the plane and take intersection of these axes as origin, represented by $O$ and every other point is defined with reference to this point. This method gives us cartesian coordinates. The other way is choosing an origin and one axis. Every point is then defined in terms of its distance from origin and the angle made by the line joining the point and origin with the axis. This method gives us Polar Coordiantes. We will first study cartesian coorsinates, and later we will study polar coordinates.

1.3. Cartesian Coordinate System

In general, when we study cartesian coordinate system we mamke use of two perpendicular axes. However, to prevent the loss of generality let us make use of inclined axes or oblique axes. Consider the diagram given below:

$X O X^{'}$ is called the $x$ -axis and $Y O Y^{'}$ is called the $y$ -axis. As said the point $O$ is called the origin of coordinates or simply, the origin.

Consider a point $P_{1}$ . Draw a line from it parallel to $O Y$ to meet $O X$ at $M_{1}$ . The distance $O M_{1}$ is called the abscissa and the distance $M_{1} P$ is called the ordinate of the point $P_{1}$ , while together they are called its coordinate.

Distances when measured parallel to $x$ -axis are typically denoted by $x$ with or without a suffix, for example: $x_{1}, x_{2}, \dots, x^{'}, x^{″}, \dots$ and distances when measured parallel to $y$ -axis are typically denoted by $y$ with or without a suffix, for example: $y_{1}, y_{2}, \dots, y^{'}, y^{″}, \dots$ .

Figure 1.2. Cartesian axes in xy-plane.

A point $P$ having the abscissa $x$ and the ordinate $y$ is typically denoted as $(x, y)$ . For example, if a point is at two units distance from $y$ -axis and at a distance of three units from $x$ -axis then it is written as $(2, 3)$ .

Distances measured parallel to $O X$ are positive while distances measured parallel to $O X^{'}$ are negative. Similarly, distances measured parallel to $O Y$ are positive and distances parallel to $O Y^{'}$ are negative. Thus, cartesian plane is divided into four quadrants. $X O Y$ is the first quadrant in which both abscissa and ordinates are positive. $X O Y^{'}$ represents the second quadrant in which abscissa is negative and ordinate is positive. $X^{'} O Y^{'}$ is the third quadrant where both abscissa and ordinate are negative and finally $X O Y^{'}$ is the fourth quadrant in which abscissa is positive and ordinate is negative. Observe that these quadrants are taken in anti-clockwise order and similar to qudrants in trigonometry.

In Figure 1.1, “Number line” we can say that $P_{1}$ lies in first quadrant, $P_{2}$ lies in second quadrant and so on.

The axes, when they are not at right angles, are called Oblique Axes. The angle between positive $x$ -axis and $y$ -axis i.e. angle between $O X$ and $O Y$ or the $∠ X O Y$ is generally denoted by the Greek letter $Ω$ .

In general, it is convenient to take the axes at right angles. The axes are then called the Rectangular Axes. You can assume that axes are rectangular unless otherwise stated. As you can guess this system is called Cartesian Coordinate System in the honor of Des Cartes.

1.4. Distance Between Two Points

Let $P_{1}$ and $P_{2}$ be two points as shown in the diagram with coordinates $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . $M_{1} P_{1} ∥ O Y, M_{2} P_{2} ∥ O Y$ and $P_{2} R ∥ O X$ . Thus, $P_{1} R = M_{2} M_{1} = O M_{2} - O M_{1} = x_{2} - x_{1}, R P_{2} = M_{2} P_{2} - M_{1} P_{1} = y_{2} - y_{1}$ and $∠ P_{1} R P_{2} = ∠ O M_{1} P_{1} = 180^{\circ} - P_{2} M_{2} X = 180^{\circ} - ω$ .

From trigonometry, we know that $P_{1} P_{2}^{2} = P_{2} R^{2} + R P_{1}^{2} - 2 P_{2} R . R P_{1} . \cos (180^{\circ} - ω)$ which is $(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + 2 (x_{1} - x_{2}) (y_{1} - y_{2}) \cos ω$ (1.1)

In the case when axes are rectangular i.e. axes are at right angle or $ω = 90^{\circ}$ then the Eq. 1.1 reduces to $P_{1} P_{2}^{2} = (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}$ and therefore in rectangular coordinate system the distance between two points is $\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$ .

Figure 1.3. Distance between two points.

Clearly, the distance between any point $(x, y)$ and the origin $(0, 0)$ will be $\sqrt{x^{2} + y^{2}}$ .

1.5. Section Formula

Consider the diagram given below.

Figure 1.4. Section Formula.

Consider points $P_{1} (x_{1}, y_{1})$ and $P_{2} (x_{2}, y_{2})$ . The point $P (x, y)$ divides the line segment $P_{1} P_{2}$ such that $P_{1} P : P P_{2} = m_{1} : m_{2}$ then we need to find the coordinates of $P$ .

We have $O M = x_{1}, M_{1} P_{1} = y_{1}, O M = x, M P = y, O M_{2} = x_{2}$ , and $M_{2} P_{2} = y_{2}$ .

$P_{1} R_{1}, P R_{2} ∥ O X$ . Then

$P_{1} R_{1} = M M_{1} = O M - O M_{1} = x - x_{1}$

$P R_{2} = M M_{2} = O M_{2} - O M = x_{2} - x$

$R_{1} P = M P - M_{1} P_{1} = y - y_{1}$ and $R_{2} P_{2} = M_{2} P_{2} - M P = y_{2} - y$ .

From the similar triangles $△ P_{1} R_{1} P$ and $△ P R_{2} P_{2}$ we have

$\frac{m_{1}}{m_{2}} = \frac{P_{1} P}{P P_{2}} = \frac{P_{1} R_{1}}{P R_{2}} = \frac{x - x_{1}}{x_{2} - x}$

$∴ x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}$ . Similarly, it can be found that $∴ y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}}$ .

Thus, the coordinates of the point which divides $P_{1} P_{2}$ in the ratio $m_{1} : m_{2}$ is $(x, y) = (\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}})$ (1.2)

If the point $P$ divides these two points externally then we can prove in a similar manner that the point’s coordinates is $(x, y) = (\frac{m_{1} x_{2} - m_{2} x_{1}}{m_{1} - m_{2}}, \frac{m_{1} y_{2} - m_{2} y_{1}}{m_{1} - m_{2}})$ (1.3)

Clearly if the point is mid-point then we find that the coordinates of mid-point is given by $(x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ (1.4)

1.6. Area of a Trapezium

We will consider the area of a trapezium in this section from geometry because it will be useful in the coming sections. Consider the following diagram:

Figure 1.5. Area of a trapezium.

We know that area of a triangle is half of the product of height of any side and the perpendicular drawn from the opposite angle. Thus,

$◻ A B C D = △ A B C + △ A C D = \frac{1}{2} . B C . A L + \frac{1}{2} . A D . C M = \frac{1}{2} (B C + A D) . A L$

1.7. Area of a Triangle

Consider following diagram with $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ .

Figure 1.6. Area of a triangle.

Let $Δ$ denote the area of the $△ A B C$ . Then, $Δ A B C = ◻ A L N C + ◻ C N M B - ◻ A L M B$

$= \frac{1}{2} L N (L A + N C) + \frac{1}{2} N M (B C + M B) - \frac{1}{2} L M (L A + M B)$

$= \frac{1}{2} [(x_{3} - x_{1}) (y_{1} + y_{3}) + (x_{2} - x_{3}) (y_{2} + y_{3}) - (x_{2} - x_{1}) (y_{1} + y_{2})]$

On simplification, we obtain $Δ = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{1})]$ (1.5)

The equation can be represented in the determinant form as given below: $Δ = | \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} |$ (1.6)

In the above calculation if the axes are oblique with an angle $ω$ then the area of the triangle would be: $Δ = \frac{1}{2} \sin ω [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{1})]$ (1.7)

If one of the vertices is the origin $O (0, 0)$ then the area of the triangle would be $\frac{1}{2} (x_{1} y_{2} - x_{2} y_{1})$ .

1.7.1. Sign of Area of a Triangle

As you can see from the formula; area of a triangle could be positive or negative. There are two ways to fix this. The first is to take modulus of the equation. The second is to plot the points of triangle in clockwise direction. If we plot the points in anti-clockwise direction then area of the triangle will be negative in that case we can take modulus of the value obtained.

1.7.2. Condition of Collinearity of Three Points

From the above relation we can extrapolate that if the area of the triangle is zero then the three vertexes would collapse into a straight line. Thus the condition of collinearity of these three points can be written as: $| \begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$ (1.8)

Now that we have area of a trapezium and area of a triangle finding area of other polygons is trivial and we will find them in our exercises.

1.8. Centroid of a Triangle

Figure 1.7. Centroid of a triangle.

The point of intersection of the medians of the triangle is called the centroid of the triangle.

Let $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ be the vertices of the $△ A B C$ . Let $A D, B E$ and $C F$ be the three medians.

Since $D$ is middle point of $B D \Rightarrow D = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ .

Let $G$ be the centroid i.e. intersection of the three medians. $G$ will divide $A D$ in the ratio $2 : 1$ i.e. $A G : G D$ then

G = (\frac{2 \frac{x_{2} + x_{3}}{2} + 1. x_{1}}{2 + 1}, \frac{2 \frac{y_{2} + y_{3}}{2} + 1. y_{2}}{2 + 1}) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})

(1.9)

Similarly it can be shown that $G$ has same coordinates for other medians. Thus, $G$ lies on the same coordinates for all three medians.

Thus, $G = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$ is the centroid of the triangle.

1.9. Incenter of a Triangle

The point of intersection of the bisectors of the angles of the triangle is called the incenter of the triangle.

Figure 1.8. Incenter of a triangle.

Let $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ be the vertices of the $△ A B C$ . Let $A D, B E$ and $C F$ be the three internal bisectors of the angles $A, B$ and $C$ respectively. Let these bisectors meet at the incenter $I$ .

Since $B C$ is internal bisector of $∠ B A C$ , therefore, $\frac{B D}{D C} = \frac{B A}{A C} = c / b$

$\Rightarrow D = (\frac{b x_{2} + c x_{3}}{b + c}, \frac{b y_{2} + c y_{3}}{b + c})$

The incenter $I$ divides $A D$ internally in the ratio $b + c : a = A I : I D$ .

$\Rightarrow I = (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$

Similarly it can be shown for two other bisectors that $I$ has the same coordinate. Thus, $I = (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$ (1.10) is the incenter of the triangle.

1.10. Area of a Quadrilateral

Consider following diagram with $A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})$ and $D (x_{4}, y_{4})$ .

Figure 1.9. Incenter of a quadrilateral.

If $A, B, C, D$ are in anti-clockwise order then area of quadrilateral will be positiive but if it is clockwise then it will be negative and we will have to take modulus of it.

Area of $◻ A B C D = Δ A B C + Δ A C D$

Another way to find area of a quadrilateral is

◻ A B C D = \frac{1}{2} [| \begin{matrix} x_{1} & y_{1} \\ x_{2} & y_{2} \end{matrix} | + | \begin{matrix} x_{2} & y_{2} \\ x_{3} & y_{3} \end{matrix} | + | \begin{matrix} x_{3} & y_{3} \\ x_{4} & h y_{4} \end{matrix} | + | \begin{matrix} x_{4} & y_{4} \\ x_{1} & y_{1} \end{matrix} |]

(1.11)

1.11. Polar Coordinates

Consider a line $O X$ through a point $O$ . We call the line initial line and we call the pole or the origin.

Figure 1.10. Polar Coordinates.

As you can figure out the position of a point $P$ can be found by knowing its distance from the pole and the positive counter-clockwise angle made by the line joining the point with the initial line. The distance $O P$ is typically denoted by $r$ and the $∠ X O P$ is denoted by $θ$ . $O P$ or $r$ is called the radius vector and $θ$ is called the vectorial angle of the point.

1.12. Polar Coordinates and Cartesian Coordinates

Figure 1.11. Polar and Cartesian Coordinates.

As we can see from the diagram a point $P$ having polar coordinates $(r, θ)$ can be represented in terms of cartesian coordinate as $(r \cos θ, r \sin θ)$ . Similarly, if a point has cartesian coordinates $(x, y)$ then $r = \sqrt{x^{2} + y^{2}}$ and $θ = \tan^{- 1} \frac{y}{x}$ would represent the equivalent polar coordinates.

1.13. Problems

Find the areas of the triangles the coordinates of whose vertices are respectively:

$(1, 3), (- 7, 6)$ and $(5, - 1)$ .
$(0, 4), (3, 6)$ and $(- 8, - 2)$ .
$(5, 2), (- 9, - 3)$ and $(- 3, - 5)$ .
$(a, b + c), (a, b - c)$ and $(- a, c)$ .
$(a \cos ϕ_{1}, a \sin ϕ_{1}), (a \cos ϕ_{2}, a \sin ϕ_{2})$ and $(a \cos ϕ_{3}, a \sin ϕ_{3})$ .
$(a m_{1}^{2}, 2 a m_{1}), (a m_{2}^{2}, 2 a m_{2})$ and $(a m_{3}^{2}, 2 a m_{3})$ .

Prove by showing that the area of the triangle formed by them is zero that the following points are in a straight line:

$(1, 4), (3, - 2)$ and $(- 3, 16)$ .
$(\frac{1}{2}, 3), (- 5, 6)$ and $(- 8, 8)$ .
$(a, b + c), (b, c + a)$ and $(c, a + b)$ .
In any $△ A B C$ prove that $A B^{2} + A C^{2} = 2 (A C^{2} + A D^{2})$ , where $D$ is the middle point of $B C$ .
$A B C$ is a triangle and $D, E$ and $F$ are the middle points of the sides $B C, C A$ and $A B$ ; prove that the point which divides $A D$ internally in the ratio $2 : 1$ also divides the line $B E$ and $C F$ in the same ratio.
Find the area of a quadrilateral whose vertices are $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ and $(x_{4}, y_{4})$ .

Find the areas of the quadrilaterals the coordinates of whose vertices, taken in order, are

$(1, 1), (3, 4), (5, - 2)$ and $(4, - 7)$ .
$(- 1, 0), (- 3, - 9), (5, 8)$ and $3, 9$ .

Find the lengths of the straight lines joining the pairs of points whose polar coordinates are

$(r_{1}, θ_{1})$ and $(r_{2}, θ_{2})$
$(2, 30^{\circ})$ and $(4, 120^{\circ})$ .
$(- 3, 45^{\circ})$ and $(7, 105^{\circ})$ .
$(a, \frac{π}{2})$ and $(3 a, \frac{π}{6})$ .

Find the areas of the triangle whose coordinates are

$(r_{1}, θ_{1}), (r_{2}, θ_{2})$ and $(r_{3}, θ_{3})$ .
$(1, 30^{\circ}), (2, 60^{\circ})$ and $(3, 90^{\circ})$ .
$(- 3, 30^{\circ}), (5, 150^{\circ})$ and $(7, 210^{\circ})$ .
$(- a, \frac{π}{6}), (a, \frac{π}{2})$ and $(- 2 a, \frac{2 π}{3})$ .

Find the distance between the points:

$(a \cos α, a \sin α)$ and $(a \cos β, a \sin β)$ .
$(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ .

Change the following equations to polar coordinates:

$x^{2} + y^{2} = a^{2}$ .
$y = x \tan α$ .
$x^{2} + y^{2} = 2 a x$ .
$x^{2} - y^{2} = 2 a y$ .
$x^{2} = y^{2} (2 a - x)$ .
${(x^{2} + y^{2})}^{2} = a^{2} (x^{2} - y^{2})$ .

Change the following equations to cartesian coordinates

$r = a$ .
$θ = \tan^{- 1} m$ .
$r = a \cos θ$ .
$r^{2} = a^{2} \sin 2 θ$ .
$r^{2} \sin 2 θ = 2 a^{2}$ .
$\sqrt{r} \cos \frac{θ}{2} = \sqrt{a}$ .
$\sqrt{r} = \sqrt{a} \sin \frac{θ}{2}$ .
$r (\cos 3 θ + \sin 3 θ) = 5 k \sin θ \cos θ$ .
Find $a$ if the distance between $(a, 2)$ and $(3, 4)$ is $8$ .
Prove that the distance between the points $(a + r \cos θ, b + r \sin θ)$ and $(a, b)$ is independent of $θ$ .
Use distance formula to show that the points $(c o s e c^{2} θ, 0), (0, \sec^{2} θ)$ and $(1, 1)$ are collinear.
If the point $P (x, y)$ be equidistant from the points $(a + b, b - a)$ and $(a - b, a + b)$ , prove that $\frac{a - b}{a + b} = \frac{x - y}{x + y}$ .
Prove that the points $(3, 4), (8, - 6)$ and $(13, 9)$ are the vertices of a right angle triangle.
Determine the type(isosceles, right angled, right angled isosceles, equilateral, scalene) of the following triangles whose vertices are
1. $(1, 1), (- \sqrt{3}, \sqrt{3})$ and $(- 1, - 1)$ .
2. $(0, 2), (7, 0)$ and $(2, 5)$ .
3. $(- 2, 5), (7, 10)$ and $(3, - 4)$ .
Prove that the distance of the point $(a \cos α, a \sin α)$ from the origin is independent of $α$ .
Let $A (6, - 1), B (1, 3)$ and $C (x, 8)$ be three points such that $A B = B C$ . Find the value of $x$ .
Using distance formula, show that the points $(1, 5), (2, 4)$ and $(3, 3)$ are collinear.
Prove that the points $(2 a, 4 a), (2 a, 6 a)$ and $(2 a + \sqrt{3 a}, 5 a)$ are the vertices of an equilateral triangle.
If the line segment joining the points $A (a, b)$ and $B (c, d)$ subtend an angle $θ$ at the origin $O$ , prove that $\cos θ = \frac{a c + b d}{\sqrt{(a^{2} + b^{2}) (c^{2} + d^{2})}}$ or $O A . O B . \cos θ = a c + b d$ .
Find the circumcenter of the triangle whose vertices are $(- 2, - 3), (- 1, 0)$ and $(7, - 6)$ . Also find the radius of the circumcircle.
The distance between two parallel lines is unity. A point $P$ lies between the lines at a distance $a$ from one of them. Find the length of a side of an equilateral $△ P Q R$ , vertex $Q$ of which lies on one of the parallel lines and vertex $R$ on the other line.
The opposite angular points of a square are $(3, 4)$ and $(1, - 1)$ , find the coordinate of the remaining vertices of the square.
$A (- 4, 0)$ and $B (- 1, 4)$ are two given points. $𝐶$ and $D$ are symmetric to the given points $A$ and $B$ respectively with respect to $y$ -axis. Calculate the perimeter of the trapezium $A B C D$ .
Point $B$ is symmetric to $A (4, - 1)$ with respect to the bisector of the first quadrant. Find $A B$ .
A line segment $A B$ through a point $A (2, 0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$ -axis is rotated about $A$ in anti-clockwise direction though an angle of $15^{\circ}$ . If $A$ be the new position of the point $B (2 + \sqrt{3}, 1)$ find the coordinate of $C$ .
The point $(1, - 2)$ is reflected in the $x$ -axis and then translated parallel to the positive direction of $𝑥$ -axis through a distance of $3$ units. Find the coordinates of the point in the new position.
The line segment joining $A (3, 0)$ and $B (5, 2)$ is rotated about $A$ in the anticlockwise direction by an angle of $45^{\circ}$ so that point $B$ goes to C. If $D$ is the reflection of $C$ in $y$ -axis, find the coordinates of $D$ .
Find the coordinates of the point which divides the line segment joining the points $(5, - 2)$ and $(9, 6)$ internally and externally in the ratio $3 : 1$ .
$𝑥$ coordinates of two points $B$ and $𝐶$ are the roots of the equation $x^{2} + 4 x + 3 = 0$ and their $𝑦$ coordinates are the roots of the equation $x^{2} - x - 6 = 0$ . If $x$ coordinates of $𝐵$ is less than $𝐵$ coordinate of $C$ and $y$ coordinate of $B$ is greater than the $𝐵$ coordinate of $C$ and coordinate of a third point $𝐴$ be $(3, - 5)$ , find the length of the bisector of the interior angle at $A$ .
Find the ratio in which the point $(2, y)$ divides the line segment joining $(4, 3)$ and $(6, 3)$ and hence find the value of $y$ .
If $A, B, C, D$ are points whose coordinates are $(- 2, 3), (8, 9), (0, 4), (3, 0)$ respectively, find the ratio in which $A B$ is divided by $C D$ .
If $A_{1}, A_{2}, A_{3}, \dots, A_{n}$ are $n$ points in a plane whose coordinates are $(x_{1}, y_{1}),$ $(x_{2}, y_{2}), (x_{3}, y_{3}), \dots, (x_{n}, y_{n})$ respectively. $A_{1} A_{2}$ is bisected in the point $G_{1}$ ; $G_{1} A_{3}$ is divided at $G_{2}$ in the ratio $1 : 2$ ; $G_{2} A_{4}$ is divided at $G_{3}$ in the ratio $1 : 3$ ; and so on until all the points are exhausted. Show that the coordinates of the final point so obtained are $\frac{x_{1} + x_{2} + \dots + x_{n}}{n}, \frac{y_{1}, y_{2} + \dots + y_{n}}{n} .$
Show that the straight line $a x + b y + c = 0$ divides the join of points $A (x_{1}, y_{1})$ and $B (x_{2}, y - 2)$ in the ratio $- \frac{a x_{1} + b y_{1} + c}{a x_{2} + b y_{2} + c}$ . Explain the negative sign.
A line $L$ intersects three sides $B C, C A$ and $A B$ of $△ A B C$ in $P, Q$ and $R$ respectively. Show that $\frac{B P}{P C} . \frac{C Q}{Q A} . \frac{A R}{R B} = - 1$ .
The vertices of a triangle are $A (x - 1, x_{1} \tan α), B (x_{2}, x_{2} \tan β), C (x_{3}, x_{3} \tan γ)$ . If the circumcenter of $△ A B C$ coincides with the origin and $H (x, y)$ is the orthocenter of $△ A B C$ , (where $x_{1} \sec α, x_{2} \sec β, x_{3} \sec γ$ are of the same sign). Show that $\frac{\overset{―}{y}}{\overset{―}{x}} = \frac{\sin α + \sin β + \sin γ}{\cos α + \cos β + \cos γ}$ .
If $α, β$ and $γ$ are the real roots of the equation $x^{3} - 3 p x^{2} + 3 q x - 1 = 0$ , find the centroid of the triangle whose vertices are $(α, \frac{1}{α}), (β, \frac{1}{β})$ and $(γ, \frac{1}{γ})$ .
If $A (a t^{2}, 2 a t), B (\frac{a}{t^{2}}, - \frac{2 a}{t})$ and $C (a, 0)$ be any three points, show that $\frac{1}{A C} + \frac{1}{B C}$ is independent of $t$ .
If two vertices of an equilateral triangle be $(0, 0)$ and $(3, \sqrt{3})$ , find the coordinate of the third vertex.
Find the circumcenter and circumradius of the triangle whose vertices are $(- 2, 3), (2, - 1)$ and $(4, 0)$ .
The vertices of a triangle are $A (1, 1), B (4, 5)$ and $C (6, 13)$ . Find $\cos A$ .
Find the distance between the points $(3, \frac{π}{4})$ and $(7, \frac{5 π}{4})$ .
$A (2, 4)$ and $B (2, 6)$ are two given points; $A B P$ is an equilateral triangle on the side of $A B$ opposite to the origin. Find the coordinates of $P$ .
Show that the points $(2, 45^{\circ}), (\sqrt{2}, 90^{\circ})$ and $(- 2, 135^{\circ})$ are the vertices of a right angled triangle.
Find the coordinates of the point which divides the line segment joining $(2, 4)$ and $(6, 8)$ in the ratio $1 : 3$ internally and externally.
Find the coordinates of the points which trisect the line segment joining the points
1. $(1, - 2)$ and $(- 3, 4)$ .
2. $(2, 3)$ and $(6, 5)$ .
$A (1, 1)$ and $B (2, - 3)$ are two points and D is a point on $A B$ produced such that $A D = 3 A b$ , then find the coordinates of $D$ .
If the middle point of the line segment joining $(3, 4)$ and $(k, 7)$ is $(x, y)$ and $2 x + 2 y + 1 = 0$ , find the value of $k$ .
One end of a diameter of a circle is at $(2, 3)$ and the center is $(- 2, 5)$ , find the coordinates of the other end of the diameter.
Find the length of the medians of the triangle whose vertices are $(- 1, 3), (1, - 1)$ and $(5, 1)$ .
If the point $C (- 1, 2)$ divides the line segment joining $A (2, 5)$ and $B$ in the ratio $3 : 4$ , find the coordinates of $B$ .
$A, B, C$ are collinear points and $B$ lies between $A$ and $C$ . $A$ and $B$ are $(3, 4)$ and $(7, 7)$ respectively. If $A C = 10$ units, find the coordinates of $𝐶$ .
Find the ratio in which $(- 8, 3)$ divides the line segment of the points $(2, - 2)$ and $(- 4, 1)$ .
In what ratio does the $x$ -axis divides the line segment joining the points $(2, - 3)$ and $(5, 6)$ .
Show that the straight lines joining the points $A (0, - 1)$ and $B (15, 2)$ divides the line joining the points $C (- 1, 2)$ and $D (4, - 5)$ internally in the ratio $2 : 3$ .
Find the ratio in which the line segment joining the points $(1, 2)$ and $(- 2, 3)$ is divided by the line $3 x + 4 y = 7$ .
Find the ratio in which the line $y - x + 2 = 0$ divides the line segment joining $(3, - 1)$ and $(8, 9)$ .
Find the distance of the point from origin which divides the line segment joining the points $(5, - 4)$ and $(3, - 2)$ in the ratio $4 : 3$ .
The coordinates of the middle points of the sides of a triangle are $(1, 1), (2, 3)$ and $(4, 1)$ , find the coordinates of the vertices.
Find the centroid and incenter of the triangle whose vertices are
1. $(2, 4), (6, 4)$ and $(2, 0)$ .
2. $(1, 2), (2, 3)$ and $(3, 4)$ .
Two vertices of a triangle are $(- 1, 4)$ and $(5, 2)$ . If its centroid is $(0, - 3)$ , find the third vertex.
$A (1, 4)$ and $B (4, 8)$ are two points. $P$ is a point on $A B$ such that $A P = A B + B P$ . If $A P = 10$ , find the coordinates of $P$ .
Find the area of the triangle whose vertices $A, B, C$ are respectively $(3, 4)$ , $(- 4, 3)$ and $(8, 6)$ .
Find the area of the quadrilateral whose vertices are $(- 3, 2), (7, - 6), (- 5, - 4)$ and $(5, 4)$ .
The coordinates of points $A, B, C$ and $P$ are $(6, 3), (- 3, 5), (4, - 2)$ and $(x, y)$ respectively, prove that $\frac{Δ P B C}{Δ A B C} = \frac{| x + y - 2 |}{7}$ .
Show that the points $(3, 3), (h, 0)$ and $(0, k)$ are collinear if $\frac{1}{h} + \frac{1}{k} = \frac{1}{3}$ .
If $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ are the vertices of a $△ A B C$ and $(x, y)$ be a point on the internal bisector of $∠ A$ , then prove that $b | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | + c | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$ , where $A C = b$ and $A B = c$ .
If the points $(\frac{a^{3}}{a -}, \frac{a^{2} - 3}{a - 1}), (\frac{b^{3}}{b - 1}, \frac{b^{2} - 3}{b - 1})$ and $(\frac{c^{3}}{c - 1}, \frac{c^{2} - 3}{c - 1})$ are collinear for three distinct values $a, b$ and $c$ then show that $a b c - (a b + b c + c a) + 3 (a + b + c) = 0$ .
If $(1, 4)$ be the center of gravity of a triangle and the coordinates of its any two vertices be $(4, - 8)$ and $(- 9, 7)$ , find the area of the triangle.
Prove that the coordinates of the vertices of an equilateral triangle cannot all be rational.
If $A, B, C$ are the points $(- 1, 5), (3, 1), (5, 7)$ respectively and $D, E, F$ are the middle points of $B C, C A, A b$ respectively prove that $Δ A B C = 4 Δ D E F$ .
The vertices of a $△ A B C$ are $A (3, 0), B (0, 6)$ and $C (6, 9)$ . A straight line $D E$ divides $A B$ and $A C$ in the ratio $1 : 2$ at $D$ and E respectively, prove that $Δ A B C = 9 Δ A D E$ .
If $(t, t - 2), (t + 3, t)$ and $(t + 2, t + 2)$ are the vertices of a triangle, show that its area if independent of $t$ .
If $A (x, y), B (1, 2)$ and $C (2, 1)$ are the vertices of a triangle of area $6$ units, show that $x + y = 15$ or $- 9$ .
Find the area of the quadrilateral whose vertices are $(1, 1), (7, 3), (12, 2)$ and $(7, 21)$ .
Find the area of the pentagon whose vertices are $(4, 3), (- 5, 6), (0, - 7), (3, - 6)$ and $(- 7, - 2)$ .
Find the area of the hexagon whose consecutive vertices are $(5, 0), (4, 2), (1, 3),$ $(- 2, 2), (- 3, - 1)$ and $(0, - 4)$ .
Find the area of the triangle whose vertices are $((a + 1) (a + 2), a + 2), ((a + 2) (a + 3), a + 3)$ and $((a + 3) (a + 4), a + 4)$ .
The point $A$ divides the join of $P (- 5, 1)$ and $Q (3, 5)$ in the ratio $k : 1$ . Find the two values of k for which the area of $△ A B C$ , where $B (1, 5)$ and $C (7, - 2)$ are given, is equal to $2$ units in magnitude.
The coordinates of $A, B, C, D$ are $(6, 3), (- 3, 5), (4, - 2), (x, 3 x)$ respectively. If $Δ A B C = Δ B C D$ , find $x$ .
If the area of the quadrilateral whose angular points taken in order are $(1, 2), (- 5, 6), (7, - 4)$ and $(h, - 2)$ be zero; show that $h = 3$ .
Find the area of the triangle whose vertices $A, B, C$ are $(3, 4), (- 4, 3), (8, 6)$ respectively, and hence, find the length of perpendicular from $A$ on $B C$ .
The coordinates of the centroid of a triangle and those of two of its vertices are respectively $(\frac{2}{3}, 2), (2, 3), (- 1, 2)$ . Find the area of the triangle.
The area of a triangle is $3$ square units. Two of its vertices are $A (3, 1), B (1, - 3)$ and the centroid off the triangle lies on x-axis. Find the coordinates of the third vertex is $C$ .
$A$ and $B$ are the points $(3, 4)$ and $(5, - 2)$ , find the point $P$ such that $P A + P B$ and $Δ P A B = 10$ .
If the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ be collinear, show that $\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0$ .
f the points $(a, b), (a_{1}, b_{1})$ and $(a - a_{1}, b - b_{1})$ are collinear show that $\frac{a}{a_{1}} = \frac{b}{b_{1}}$ .
Show that the points $(a, 0), (0, b)$ and $(1, 1)$ are collinear if $\frac{1}{a} + \frac{1}{b} = 1$ .
Prove that the points $(- 4, - 1), (- 2, - 4), (4, 0)$ and $(2, 3)$ are the vertices of a rectangle.
If the points $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3},_{3})$ be the three consecutive vertices of a parallelogram, find the coordinates of the fourth vertex.
In any $△ A B C$ , prove that $A B^{2} + A C^{2} = 2 (A C^{2} + B C^{2})$ , where $D$ is the middle point of $B C$ .
If $G$ be the centroid of the $△ A B C$ and $O$ be any other point in the plane of the $A B C$ , then prove that $O A^{2} + O B^{2} + O C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 G O^{2}$ .
Prove that the area of a triangle is four times the area of the triangle formed by joining the mid points of its sides.
Prove that the line segment joining the middle points of two sides of a triangle is half the third side.
If $P, Q, R$ divide the sides $B C, C A, A B$ of $△ A B C$ in the same ratio, prove that the centroid of both the triangles coincide.
Prove that in any triangle four times the sum of the squares of the medians is equal to three times the sum of the squares of the sides.
If $G$ be the centroid of a $△ A B C$ , prove that $A B^{2} + B C^{2} + C A^{2} = 3 (G A^{2} + G B^{2}$ $+ G C^{2})$ .
Show that the line joining the centroid of a triangle to its vertices divide it into three triangles of equal area.
Show that the middle point of the hypotenuse of a right angled triangle is equidistant from its vertices.
$A (a_{1}, b_{1}), B (a_{2}, b_{2}), C (a_{3}, b_{3})$ are the vertices of a $△ A B C$ . The side $A B$ is divided by the point $D$ in the ratio $λ : μ$ and then the line segment $D C$ is divided by the point $E$ in the ratio $μ : λ + μ$ . Find the coordinates of $E$ .
The four points $A (α, 0), B (β, 0), C (γ, 0)$ and $D (δ, 0)$ are such that $a l p h a, b e t a$ are the roots of the equation $a x^{2} + 2 b x + b = 0$ and $γ, δ$ are those of the equation $a x^{2} + 2 b^{'} x + b^{'} = 0$ . Show that the sum of the ratios in which $C$ and $D$ divide $A B$ is zero if $a b^{'} + a^{'} b = 2 b b^{'}$ .
$A (1, - 2)$ and $B (2, 5)$ are two points. The lines $O A, O B$ are produced to $C$ and $D$ respectively such that $O C = 2 O A$ and $O D = 2 O B$ . Find $C D$ .
Two vertices of a triangle are $A (2, 1)$ and $B (3, - 1)$ . The third vertex $C$ lies on the line $y = x + 9$ . If the centroid of $△ A B C$ lies on the $y$ -axis, find the coordinates of $C$ and the centroid.
If $(α, β), (x, y)$ and $(p, q)$ are the coordinates of the circumcenter, the centroid and the orthocenter of a triangle, prove that $3 x = 2 α + p$ and $3 y = 2 β + q$ .
Find the coordinates of the centroid, circumcenter and orthocenter of the triangle whose vertices are $(2, 3), (3, 4)$ and $(6, 8)$ .
If $A (α, \frac{1}{α}), B (β, \frac{1}{β})$ and $C (γ, \frac{1}{γ})$ be the vertices of a $△ A B C$ , where $α, β$ are the roots of the equation $x^{2} - 6 p_{1} x + 2 = 0$ ; $β, γ$ are the roots of the equation $x^{2} - 6 p_{2} x + 3 = 0$ and $γ, α$ are the roots of the equation $𝑥 x^{2} - 6 p_{3} x + 6 = 0$ , $p_{1}, p_{2}, p_{3}$ being positive, find $p_{1}, p_{2}, p_{3}$ and the coordinates ofo the centroid of $△ A B C$ .
If $\tan α, \tan β, \tan γ$ are the roots of the equation $x^{3} - 3 a x^{2} + 3 b x - 1 = 0$ , find the centroid of the triangle whose vertices are $(\tan α, \cot α), (\tan β, \cot β)$ and $(\tan γ, \cot γ)$ .
Two unlike forces equal to $30$ and $40$ newtons are applied at the point $A (- 3, - 1)$ and $B (4, 6)$ respectively. Find the point of application of resultant force.
The area of a parallelogram is $12$ units. Two of its vertices are the points $A (- 1, 3)$ and $B (- 2, 4)$ . Find the other two vertices of the parallelogram if the point of intersection of diagonals lies on the positive side of $x$ -axis.
Give the points $A (1, 2), B (8, 4), C (4, 10)$ find the coordinates of the point $P$ such that the triangles $P C B, P C A$ and $P A B$ have the same area in magnitude and sign.
If $a, b, c$ be the $p$ th, $q$ th and $r$ th terms respectively of an H.P., show that the points $(b c, p), (c a, q), (a b, r)$ are collinear.
If $x_{1}, x_{2}, x_{3}$ are in A.P. and $y_{1}, y_{2}, y_{3}$ are also in A.P. prove that the points $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ are collinear.
If $a, b, c$ are distinct real numbers, show that the points $(a, a^{2}), (b, b^{2})$ and $(c, c^{2})$ are not collinear.
If $A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})$ are the vertices of a $△ A B C$ and $(x, y)$ be a point on the median through $A$ , show that $| \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | + | \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{3} & y_{3} & 1 \end{matrix} | = 0$ .
he area of a triangle is $\frac{3}{2}$ sq. units. Two of its vertices are $A (2, - 3)$ and $B (3, - 2)$ , the centroid of the triangle lies on the line $3 x - y - 8 = 0$ . Find the third vertex $C$ .
rove that the quadrilateral whose vertices are $A (- 2, 5), B (4, - 1), C (9, 1)$ and $D (3, 7)$ is a parallelogram and find its area. If $E$ divides $A C$ in the ratio $2 : 1$ , prove that $D, E$ and the middle point $F$ of $B C$ are collinear.
Prove that points $(- 3, - 1), (2, - 1), (1, 1)$ and $(- 2, 1)$ taken in order are vertices of a trapezium.
If the vertices of a triangle have integral coordinates, prove that the triangle cannot be equilateral.

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