Intercept Form of Straight Line Formula | Solved Examples | Practice Problems, FAQs, Important Questions

Class 11 Mathematics | Written by Neeraj Anand

Published by ANAND TECHNICAL PUBLISHERS

Introduction

The intercept form of a straight line is a special way of expressing the equation of a line in terms of its intercepts on the x-axis and y-axis. This form is particularly useful in coordinate geometry for quickly determining how a line interacts with the coordinate axes.

Intercept Form of a Straight Line

We can derive the equation of a straight line in intercept form in two ways. Let’s derive the formula of the intercept form of linear equation.

Suppose a line L makes x-intercept a and y-intercept b on the axes. That means L meets x-axis at the point (a, 0) and y-axis at the point (0, b), respectively as shown in the below figure.

As we know, the equation of a line in two point form is:

\(\begin{array}{l}\large {y-y_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})\end{array} \)

Here,

(x₁, y₁) = (a, 0)

(x₂, y₂) = (0, b)

Substituting these coordinates in the above formula,

\(\begin{array}{l}y – 0=\frac{b-0}{0-a}(x-a)\end{array} \)

y = (-b/a) (x – a)

ay = -bx + ab

bx + ay = ab

Dividing by “ab” on both sides,

(bx/ab) + (ay/ab) = ab/ab

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

Therefore, the equation of the line making intercepts a and b on x-and y-axis, respectively, is given as:

\(\begin{array}{l}\frac{x}{a}+\frac{y}{b}=1\end{array} \)

Intercept form Formula

The equation for a straight line in intercept form is given by the formula,

\(\begin{array}{l}\large \frac{x}{a}+\frac{y}{b}=1\end{array} \)

Where,

a = x-intercept

b = y-intercept

This formula can also be derived using the general equation of a line.

Intercept Form Solved Examples

Example 1: Find the equation of the line, which makes intercepts –4 and 5 on the x- and y-axes, respectively.

Solution:

Given,

x-intercept = a = -4

y-intercept = b = 5

Equation of the line in intercept form is:

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

Substituting the values of a and b, we get the equation as:

(x/-4) + (y/5) = 1

Or

(-5x + 4y)/20 = 1

-5x + 4y = 20

5x – 4y + 20 = 0

Example 2: Write the intercepts of the straight line represented by the equation 2x – 3y + 6 = 0 on the coordinate axes.

Solution:

Given line equation is:

2x – 3y + 6 = 0

2x – 3y = -6

Dividing both sides of the equation by -6,

(2x – 3y)/(-6) = (-6)/(-6)

(2x/-6) – (3y/-6) = 1

(x/-3) + (y/2) = 1

This is of the form (x/a) + (y/a) = 1.

So, a = -3 and b = 2

Therefore, the x-intercept is -3 and y-intercept is 2 for the given equation of a line.

Example 3: Let two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis, respectively. Find the equation of the line.

Solution: Given, two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis.

From the equation of the line we know,

x/a + y/b = 1 ……….. (1)

Here, a = 2 and b = 3

Therefore, putting the values of intercepts a and b, in equation 1, we get:

=>x/2 + y/3 = 1

=> 3x + 2y = 6

=> 3x + 2y – 6 = 0,

Therefore, the equation of the line is 3x + 2y – 6 = 0.

Example 4: Find the equation of the line, which makes intercepts –3 and 2 on the x- and y-axes respectively.

Solution: Given, a = –3 and b = 2.

By intercept form, we know that;

x/a + y/b = 1

x/-3 + y/2 = 1

Or

2x – 3y + 6 = 0.

Hence, this is the required equation.

Example 5: A line passes through P (1, 2) such that its intercept between the axes is bisected at P. What is the equation of the line?

Solution: The equation of a line making intercepts a and b with x-axis and y-axis, respectively, is given by:

x/a + y/b = 1

1 = (a+0)/2 ⇒ a = 2

2 = (0 + b)/2 ⇒ b = 4

Therefore, the required equation of line is;

x/2 + y/4 = 1

⇒ 2x + y – 4 = 0

Example 6: The equation of a line is given by, 2x – 6y +3 = 0. Find the slope and both the intercepts.
Solution:

The given equation 2x – 6y + 3 = 0 can be represented in slope-intercept form as:

y = x/3 + 1/2

Comparing it with y = mx + c,
Slope of the line, m = 1/3

Also, the above equation can be re-framed in intercept form as;

x/a + y/b = 1

2x – 6y = -3

x/(-3/2) – y/(-1/2) = 1

Thus, x-intercept is given as a = -3/2 and y-intercept as b = 1/2.

Example 7: The equation of a line is given by, 13x – y + 12 = 0. Find the slope and both the intercepts.

Solution: The given equation 13x – y + 12 = 0 can be represented in slope-intercept form as:

y = 13x + 12
Comparing it with y = mx + c,
Slope of the line, m = 13

Also, the above equation can be re-framed in intercept form as;

x/a + y/b = 1

13x – y = -12

x/(-12/13) + y/12 = 0

Thus, x-intercept is given as a = -12/13 and y-intercept as b = 12.

Practice Problems

1. Find the x-intercept and y-intercept for the line 5x – 8y = 2.
2. If the y-intercept of a line is -4 and the slope is 2/3, then write its equation.
3. What is the equation of a line whose x and y-intercepts are given as 1/3 and -3?

Conclusion

The intercept form of a straight line is a crucial concept in Class 11 Mathematics and is widely used in CBSE Board Exams and JEE Mains/Advanced. Understanding this form simplifies solving problems related to straight lines, making it an essential tool for students.

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