# Concept: Coordinate Geometry

CONTENTS

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 = (x1, y1) and B = (x2, y2) 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 = (x1, y1) 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 L1: ax1 + by1 + c1 = 0 and L2: ax2 + by2 + c2 = 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 = (x1, y1) and B = (x2, y2) 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 = (x1, y1) and B = (x2, y2) 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 = (x1, y1) 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.,
m1 = m2

If L1 is ax + by + c = 0, then equation of a line L2 which is parallel to L1 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.,
m1 × m2 = -1

If L1 is ax + by + c = 0, then equation of a line L2 which is perpendicular to L1 can be written as bx - ay + c' = 0

Remember:
• 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 = (x1, y1) and B = (x2, y2) 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 = (x1, y1) and B = (x2, y2) 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 = (x1, y1); B = (x2, y2) and C = (x3, y3) is given by

Area = ½ × (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))

CENTROID OF A TRIANGLE

Centroid of a triangle whose vertices are A = (x1, y1); B = (x2, y2) and C = (x3, y3) 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:
x2 + y2 = r2

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 = r2

General Equation of a Cricle

The general equation of a circle is: x2 + y2 + 2gx + 2fy + c = 0

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

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