# Concept: Coordinate Geometry

CONTENTS

- Introduction
- Distance
- Line
- Slope of a Line
- Intercepts of a Line
- Equation of a Line
- General Form of Equation of a Line
- 2-Point Form of Equation of a Line
- Point-Slope of Equation of a Line
- Slope-Intercept Form of Equation of a Line
- Intercept Form of Equation of a Line
- Parallel and Perpendicular Lines
- Division of a line segment
- Triangle
- Circle

**INTRODUCTION**

Coordinate Geometry may not be an important topic from CAT point of view, but you may expect questions from this topic in OMETs. The topic is fairly easy and questions in exam are limited to 2 dimensions.

**DISTANCE**

**DISTANCE BETWEEN 2 POINTS**

If A = (x_{1}, y_{1}) and B = (x_{2}, y_{2}) are two points in x-y plane, the distance between them is

${\mathrm{D}}_{\mathrm{AB}}=\sqrt{{\left({\mathrm{x}}_{1}-{\mathrm{x}}_{2}\right)}^{2}+{\left({\mathrm{y}}_{1}-{\mathrm{y}}_{2}\right)}^{2}}$

**DISTANCE BETWEEN A POINT AND A LINE**

Perpendicular distance of a point A = (x_{1}, y_{1}) from a line L: ax + by + c = 0 is given by

${\mathrm{D}}_{\mathrm{AL}}=\left|\frac{{\mathrm{ax}}_{1}+{\mathrm{by}}_{1}+\mathrm{c}}{\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}}\right|$

**DISTANCE BETWEEN 2 LINES**

Distance between two parallel lines L_{1}: ax_{1} + by_{1} + c_{1} = 0 and L_{2}: ax_{2} + by_{2} + c_{2} = 0 is given by

$\mathrm{D}=\left|\frac{{\mathrm{c}}_{1}-{\mathrm{c}}_{2}}{\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}}\right|$

**LINE**

A two variable equation i.e., ax + by + c = 0 represents a straight line on the x-y plane

**SLOPE OF A LINE**

A line joining two points A = (x_{1}, y_{1}) and B = (x_{2}, y_{2}) is given by

${\mathrm{S}}_{\mathrm{AB}}=\frac{{\mathrm{y}}_{2}-{\mathrm{y}}_{1}}{{\mathrm{x}}_{2}-{\mathrm{x}}_{1}}$

**INTERCEPT OF A LINE**

**x-intercept** is the point where the line cuts the x-axis. We can obtain the x-intercept by putting the value of y = 0 in the equation of the line.

**y-intercept** is the point where the line cuts the y-axis. We can obtain the y-intercept by putting the value of x = 0 in the equation of the line.

**EQUATION OF A LINE**

**GENERAL FORM OF EQUATION OF A LINE**

General Form of equation of a line is given by:

ax + by + c = 0

**2-POINT FORM OF EQUATION OF A LINE**

A line passing through two points A = (x_{1}, y_{1}) and B = (x_{2}, y_{2}) is given by:

$\frac{{\mathrm{y}}_{2}-{\mathrm{y}}_{1}}{{\mathrm{x}}_{2}-{\mathrm{x}}_{1}}=\frac{\mathrm{y}-{\mathrm{y}}_{1}}{\mathrm{x}-{\mathrm{x}}_{1}}$

**POINT-SLOPE FORM OF EQUATION OF A LINE**

Equation of a line whose slope is m and passes through the point A = (x_{1}, y_{1}) is given by:

$\mathrm{m}=\frac{\mathrm{y}-{\mathrm{y}}_{1}}{\mathrm{x}-{\mathrm{x}}_{1}}$

**SLOPE-INTERCEPT FORM OF EQUATION OF A LINE**

Equation of a line whose slope is m and y-intercept is (0, c) is given by:

y = mx + c

**INTERCEPT FORM OF EQUATION OF A LINE**

Equation of a line whose x and y intercepts are a and b respectively is given by:

$\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$

**PARALLEL & PERPENDICULAR LINES**

**PARALLEL LINES**

For two lines to be parallel their slopes should be same, i.e.,

m_{1} = m_{2}

If L_{1} is ax + by + c = 0, then equation of a line L_{2} which is parallel to L_{1} can be written as
ax + by + c' = 0

**PERPENDICULAR LINES**

For two lines to be perpendicular, product of their slopes should be -1, i.e.,

m_{1} × m_{2} = -1

If L_{1} is ax + by + c = 0, then equation of a line L_{2} which is perpendicular to L_{1} can be written as
bx - ay + c' = 0

- A line parallel to x-axis is written as y = a (constant)
- A line parallel to y-axis is written as x = a (constant)

**DIVISION OF A LINE SEGMENT**

**INTERNAL DIVISION OF A LINE SEGMENT**

If a point (C) divided the line segment joining the points A = (x_{1}, y_{1}) and B = (x_{2}, y_{2}) internally
in the ratio of m : n, then coordinates of C are:

$\mathrm{C}=\left(\frac{{\mathrm{mx}}_{2}+{\mathrm{nx}}_{1}}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_{2}+{\mathrm{ny}}_{1}}{\mathrm{m}+\mathrm{n}}\right)$

**EXTERNAL DIVISION OF A LINE SEGMENT**

If a point (C) divided the line segment joining the points A = (x_{1}, y_{1}) and B = (x_{2}, y_{2}) externally
in the ratio of m : n, then coordinates of C are:

$\mathrm{C}=\left(\frac{{\mathrm{mx}}_{2}-{\mathrm{nx}}_{1}}{\mathrm{m}-\mathrm{n}},\frac{{\mathrm{my}}_{2}-{\mathrm{ny}}_{1}}{\mathrm{m}-\mathrm{n}}\right)$

**TRIANGLE**

**AREA OF A TRIANGLE**

Area of a triangle whose vertices are A = (x_{1}, y_{1}); B = (x_{2}, y_{2}) and C = (x_{3}, y_{3}) is given by

Area = ½ × (x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}))

**CENTROID OF A TRIANGLE**

Centroid of a triangle whose vertices are A = (x_{1}, y_{1}); B = (x_{2}, y_{2}) and C = (x_{3}, y_{3}) is given by

$\mathrm{G}=\left(\frac{{\mathrm{x}}_{1}+{\mathrm{x}}_{2}+{\mathrm{x}}_{3}}{3},\frac{{\mathrm{y}}_{1}+{\mathrm{y}}_{2}+{\mathrm{y}}_{3}}{3}\right)$

**CIRCLE**

**EQUATION OF A CIRCLE**

**Standard Equation of a Cricle**

Equation of a circle with origin (0, 0) as center and 'r' as radius is:

x^{2} + y^{2} = r^{2}

**Equation of a Cricle in Radius Form**

Equation of a circle with origin (h, k) as center and 'r' as radius is:

(x - h)^{2} + (y - k)^{2} = r^{2}

**General Equation of a Cricle**

The general equation of a circle is: x^{2} + y^{2} + 2gx + 2fy + c = 0

Center of circle represented by this equation is (-g, -f) and radius of this circle is $\sqrt{{\mathrm{g}}^{2}+{\mathrm{f}}^{2}-\mathrm{c}}$