# Intercept Form of a Line Equation

Here you will learn intercept form of a line equation i.e. intercept cut by line on x-axis and y-axis with examples.

Let’s begin –

## Intercept Form of a Line

The equation of a line which cuts off intercepts a and b respectively from the x and y-axes is

$$x\over a$$ + $$y\over b$$ = 1.

Proof : Let AB be the line which cuts off intercepts OA = a and OB = b on the x and y axes respectively.

Let P(x,y) be any point on the line. Draw PL $$\perp$$ OX. Then, OL = x and PL = y.

Clearly,

Area of OAB = Area of triangle OPA + Area of triangle OPB

$$1\over 2$$ OA.OB = $$1\over 2$$ OA.PL + $$1\over 2$$ OB.PM

$$1\over 2$$ab = $$1\over 2$$ay + $$1\over 2$$bx

$$\implies$$ ab = ay + ax

$$\implies$$ $$x\over a$$ + $$y\over b$$ = 1.

This is the required equation of the line in the intercept form.

Example : Find the equation of the line which cut off an intercept 4 on the positive direction of x-axis and an intercept 3 on the negative direction of y-axis.

Solution : Here a = 4, b = -3.

So, the equation of the line is

$$x\over a$$ + $$y\over b$$ = 1 or, $$x\over 4$$ + $$y\over -3$$ = 1. or 3x – 4y = 12.

Example : Find the equation of the straight line which passes through the point (4, -2) and whose intercept on y-axis is twice that on X-axis.

Solution : let the equation of line be

$$x\over a$$ + $$y\over b$$ = 1               ………….(i)

It is given that its y-intercept is twice the x-intercept

$$\therefore$$  b = 2a

Putting b = 2a in (i), we get

2x + y = 2a                       ………………..(ii)

It passes through point (4, -2).

$$\therefore$$  8 – 2 = 2a $$\implies$$ a = 3.

Substituting a = 3 in (ii), we get

2x + y = 6 as the equation of the required line.