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.intercept form

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.

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