Coordinate Geometry — Concept, Formulas & Examples

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 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:

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

 

PARALLEL & PERPENDICULAR 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

 

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

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:

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

 

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 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

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

 

CIRCLE

 

EQUATION OF A CIRCLE

 

Equation of a circle with origin (0, 0) as center and 'r' as radius is:
x² + y² = r²

 

Equation of a circle with origin (h, k) as center and 'r' as radius is:
(x - h)² + (y - k)² = r²

 

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 $\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}}$