Intercept Form: Equation, Properties, Cases & Solved Examples 

1.0Introduction

In Coordinate Geometry, there are different ways to write the equation of a line, such as slope-intercept form, point-slope form, two-point form, and intercept form.

For people who want to take the JEE, it's very important to know how to find the intercept form of a straight line. This is because it directly helps with problems that involve intersections, geometry-based reasoning, and analytical geometry.

2.0Basics of Straight Line Equations

A straight line in a 2D coordinate plane can be written in multiple ways depending on the information available:

• $\text{Slope-intercept form: }y=mx+c$
• $\text{Point-slope form: }y-y_1=m(x-x_1)$
• $\text{Two-point form: }\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$
• $\text{Intercept form: }\frac{x}{a}+\frac{y}{b}=1$

Among these, the intercept form is useful when we know where a line cuts the x-axis and y-axis.

3.0What is Intercept Form?

The Intercept Form of a line shows how the line cuts the x-axis and y-axis.

If a line cuts the x-axis at (a,0) and the y-axis at (0,b), then the equation of the line is:

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

Here:

• a = x-intercept
• b = y-intercept

This form is one of the most frequently used equations in JEE for solving geometry-based problems.

4.0Derivation of Intercept Form

Let the line cut the x-axis at (a, 0) and y-axis at (0, b).

Equation of the line through two points (a,0) and (0,b):

$\frac{y-0}{b-0}=\frac{x-a}{0-a}$

$\frac{y}{b}=\frac{x-a}{-a}$

$\frac{y}{b}=\frac{-x+a}{a}$

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

Hence proved.

5.0Standard Equation of Intercept Form

The equation of a line in intercept form is:

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

Where:

• a = intercept on x-axis
• b = intercept on y-axis

6.0Important Cases and Conditions

(i) When both intercepts are positive

The line passes through the first quadrant, cutting both axes.

(ii) When one intercept is zero

• If a=0, the line is vertical: x=0.
• If b=0, the line is horizontal: y=0.

(iii) When line passes through origin

Both intercepts become infinite, and the equation reduces to slope form: y=mx.

7.0Properties of Intercept Form

1. The intercept form always cuts both axes (except when one intercept is infinite).
2. For a > 0, b > 0, the line lies in the first quadrant.
3. The slope of the line can be calculated as: $m=-\frac{b}{a}$
4. If one intercept is negative, the line lies in a different quadrant.
5. Equation is undefined if both a and b are zero.

8.0Solved Examples on Intercept Form

Example 1: 

Find the equation of a line that cuts the x-axis at 4 and y-axis at 5.

Solution:

$\frac{x}{4}+\frac{y}{5}=1$

Example 2:

Find slope of line $\frac{x}{3}+\frac{y}{7}=1$

Solution:
Comparing with intercept form: slope $m=-\frac{b}{a}=-\frac{7}{3}$

Example 3:

Find intercepts of line 3x+4y=12.

Solution:
Divide both sides by 12:

$\frac{x}{4}+\frac{y}{3}=1$

So, x-intercept = 4, y-intercept = 3.

Example 4:

Equation of a line with equal intercepts on both axes.

Solution:
Let intercepts be a = a. Equation:

$\frac{x}{a}+\frac{y}{a}=1⟹x+y=a$

Example 5:

Find equation of line making intercepts −3 on x-axis and 2 on y-axis.

Solution:

$\frac{x}{-3}+\frac{y}{2}=1⟹-\frac{x}{3}+\frac{y}{2}=1$