Equation of Straight Line in Intercept Form

The equation of a straight line can be expressed in various forms, one of which is the intercept form. This form is particularly useful when we know the x-intercept and y-intercept of a line.

Understanding the Intercept Form

The intercept form of the equation of a straight line is given by:

[ \frac{x}{a} + \frac{y}{b} = 1 ]

where a is the x-intercept and b is the y-intercept. This means that the line crosses the x-axis at the point (a, 0) and the y-axis at the point (0, b).

Important Points

• The intercepts a and b are the distances from the origin to the points where the line intersects the axes.
• If a or b is negative, it indicates that the intercept is on the negative side of the respective axis.
• If a is infinite, the line is parallel to the x-axis, and if b is infinite, the line is parallel to the y-axis.

Differences and Important Points

Feature	Intercept Form	Standard Form	Slope-Intercept Form
General Equation	$\frac{x}{a} + \frac{y}{b} = 1$	$Ax + By + C = 0$	$y = mx + c$
Intercepts Known	Yes	No (unless further information is provided)	No (only y-intercept is known)
Slope (m)	$-\frac{a}{b}$	$-\frac{A}{B}$	Given as 'm'
X-Intercept	a	$-\frac{C}{A}$ (if B ≠ 0)	$\frac{-c}{m}$
Y-Intercept	b	$-\frac{C}{B}$ (if A ≠ 0)	c

Formulas

• Slope of the line in intercept form: $m = -\frac{a}{b}$
• Distance from a point $(x_1, y_1)$ to the line: $d = \frac{|x_1/a + y_1/b - 1|}{\sqrt{1/a^2 + 1/b^2}}$

Examples

Example 1: Finding the Equation of a Line

Suppose a line has an x-intercept of 5 and a y-intercept of 3. The equation of the line in intercept form is:

[ \frac{x}{5} + \frac{y}{3} = 1 ]

Example 2: Converting to Slope-Intercept Form

Given the equation in intercept form $\frac{x}{4} + \frac{y}{2} = 1$, we can convert it to slope-intercept form:

[ y = -\frac{1}{2}x + 2 ]

Here, the slope (m) is $-\frac{1}{2}$ and the y-intercept (c) is 2.

Example 3: Finding the Distance from a Point to the Line

Given the line $\frac{x}{6} + \frac{y}{8} = 1$ and a point (3, 4), find the distance from the point to the line.

Using the distance formula:

[ d = \frac{|3/6 + 4/8 - 1|}{\sqrt{1/6^2 + 1/8^2}} = \frac{|1/2 + 1/2 - 1|}{\sqrt{1/36 + 1/64}} = \frac{0}{\sqrt{1/36 + 1/64}} = 0 ]

The point (3, 4) lies on the line, so the distance is 0.

By understanding the intercept form of a straight line, students can solve various problems related to the geometry of lines, including finding the equation of a line given its intercepts, converting between different forms of the line equation, and calculating the distance from a point to a line.