Concept: Coordinate Geometry

CONTENTS

• Introduction
• Distance
• Distance between 2 Points
• Distance between a Point and a Line
• Distance between 2 Parallel Lines
• 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
• Parallel Lines
• Perpendicular Lines
• Division of a line segment
• Internal Division of a Line Segment
• External Division of a Line Segment
• Triangle
• Area of a triangle
• Coordinates of Centroid of a Triangle

 

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.

 

 

If A = (x₁, y₁) and B = (x₂, y₂) are two points in x-y plane, the distance between them is

${\mathrm{D}}_{\mathrm{AB}}=\sqrt{{\left({\mathrm{x}}_1-{\mathrm{x}}_2\right)}^2+{\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)}^2}$

 

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 two parallel lines L₁: ax₁ + by₁ + c₁ = 0 and L₂: ax₂ + by₂ + c₂ = 0 is given by

$\mathrm{D}=\left|\frac{{\mathrm{c}}_1-{\mathrm{c}}_2}{\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}}\right|$

 

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

 

A line joining two points A = (x₁, y₁) and B = (x₂, y₂) is given by

${\mathrm{S}}_{\mathrm{AB}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}$

 

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.

 

 

General Form of equation of a line is given by:

ax + by + c = 0

 

A line passing through two points A = (x₁, y₁) and B = (x₂, y₂) is given by:

$\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}=\frac{\mathrm{y}-{\mathrm{y}}_1}{\mathrm{x}-{\mathrm{x}}_1}$

 

Equation of a line whose slope is m and passes through the point A = (x₁, y₁) is given by:

$\mathrm{m}=\frac{\mathrm{y}-{\mathrm{y}}_1}{\mathrm{x}-{\mathrm{x}}_1}$

 

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

y = mx + c

 

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

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

 

 

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

 

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

 

 

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:

$\mathrm{C}=\left(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{\mathrm{m}+\mathrm{n}}\right)$

 

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:

$\mathrm{C}=\left(\frac{{\mathrm{mx}}_2-{\mathrm{nx}}_1}{\mathrm{m}-\mathrm{n}},\frac{{\mathrm{my}}_2-{\mathrm{ny}}_1}{\mathrm{m}-\mathrm{n}}\right)$

 

 

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 whose vertices are A = (x₁, y₁); B = (x₂, y₂) and C = (x₃, y₃) is given by

$\mathrm{G}=\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3}{3},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3}{3}\right)$