Intercept Form

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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:

$Slope-intercept form: y = m x + c$
$Point-slope form: y - y_{1} = m (x - x_{1})$
$Two-point form: \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{x - x _{1}}{x _{2} - x _{1}}$
$Intercept form: \frac{x}{a} + \frac{y}{b} = 1$

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

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

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

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:

$Slope-intercept form: y = m x + c$
$Point-slope form: y - y_{1} = m (x - x_{1})$
$Two-point form: \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{x - x _{1}}{x _{2} - x _{1}}$
$Intercept form: \frac{x}{a} + \frac{y}{b} = 1$

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

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

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

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Intercept Form: Equation, Properties, Cases & Solved Examples

1.0Introduction

2.0Basics of Straight Line Equations

3.0What is Intercept Form?

4.0Derivation of Intercept Form

5.0Standard Equation of Intercept Form

6.0Important Cases and Conditions

(i) When both intercepts are positive

(ii) When one intercept is zero

(iii) When line passes through origin

7.0Properties of Intercept Form

8.0Solved Examples on Intercept Form

Example 1:

Example 2:

Example 3:

Example 4:

Example 5:

Table of Contents

Intercept Form: Equation, Properties, Cases & Solved Examples

1.0Introduction

2.0Basics of Straight Line Equations

3.0What is Intercept Form?

4.0Derivation of Intercept Form

5.0Standard Equation of Intercept Form

6.0Important Cases and Conditions

(i) When both intercepts are positive

(ii) When one intercept is zero

(iii) When line passes through origin

7.0Properties of Intercept Form

8.0Solved Examples on Intercept Form

Example 1:

Example 2:

Example 3:

Example 4:

Example 5:

Table of Contents