INTERCEPT FORM EQUATION OF A LINE

We will find the equation of a line whose intercepts are 'a' and 'b' on the coordinate axes respectively.

Let PQ be a line meeting X axis at A and Y axis at B.

Let OA = a, OB = b. Then the coordinates of A and B are (a, 0) and (0, b) respectively. 

Therefore, the equation of the line joining A and B is

Using two-point form, equation of line joining the two points A and B is

(y - 0)/(b - 0) = (x - a)/(0 - a)

y/b = -(x - a)/a

y/b = (-x + a)/a

y/b = -x/a + 1

Hence, Intercept-Form equation of a line is 

x/a + y/b = 1

Example 1 :

If the x-intercept and y-intercept of a straight line are 2 and -4 respectively, find the general equation of the straight line.

Solution :

Given : x- intercept is 2 and y-intercept is -4.

Equation of a straight line in intercept form :

x/a  +  y/b = 1

Substitute a = 2 and b = -4.

x/2 + y/(-4) = 1

x/2 - y/4 = 1

Least common multiple of the denominators 2 and 4 is 4.

Multiply each side by 4 to get rid of the fractions.

2x - y = 4

Subtract 4 from each side.

2x - y - 4 = 0

Example 2 :

Find the intercepts made by the line 4x + 9y - 36 = 0 on the coordinate axes.

Solution :

Write the equation in intercept form.

4x + 9y - 36 = 0

Add 36 to each side.

4x + 9y = 36

Divide each side by 36.

(4x + 9y)/36 = 36/36

4x/36 + 9y/36 = 1

x/9 + y/4 = 1

x-intercept = 9

y-intercept = 4

Example 3 :

Find the equation of a line in standard form which passes through (7, 5) and makes intercepts on the axes equal in magnitude but opposite in sign.

Solution :

Let 'a' the x-intercept of the line.

Then y-intercept is '-a'.

Equation of line in intercept form :

x/a + y/b = 1

Substitute b = -a.

x/a + y/(-a) = 1

x/a - y/a = 1

(x - y)/a = 1

x - y = a ----(1)

The line is passing through the point (7, 5).

So, substitute x = 7 and y = 5.

7 - 5 = a

2 = a

Substitute a = 2 in (1).

x - y = 2

Example 4 :

A line makes positive intercepts on coordinate axes whose sum is 7 and it passes through (-3, 8) . Find the equation of line in standard form.

Solution :

Let 'a' and 'b' be the intercepts.

a + b = 7

b = 7 - a

Intercept-form equation of a line :

x/a + y/b = 1

Substitute b = 7 - a.

x/a + y/(7 - a) = 1

–3(7 – a) + 8a = a(7 – a)

-21 + 3a + 8a = 7a - a2

a2 + 4a - 21 = 0

(a - 3)(a + 7) = 0

a = 3 or a = -7

Since a is positive,

a = 3

b = 7 – a = 7 – 3 = 4

Equation of line :

x/3 + y/4 = 1

Least common multiple of the denominators 3 and 4 is 12. 

Multiply each side by 12 to get rid of the fractions.

4x + 3y = 12

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