Question

A straight line makes intercepts a and b on the x-axis and the y-axis respectively .
Find the equation of the line .​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the required equation of line is $l$ and slope of line $l$ be m.

Given that, A straight line makes intercepts a and b on the x-axis and the y-axis respectively.

So, it means line passes through the point A (a, 0) and B (0, b) respectively.

So, slope of line $l$ passes through the point A (a, 0) and B (0, b) is

$\rm \: m \: = \: \dfrac{b - 0}{0 - a}$

$\bf\implies \: \: m \: = \: - \: \dfrac{b}{a}$

Now, we know

Slope point form of a line : - Equation of line which passes through the point $\rm \: (x_1, y_1)$ and having slope m is given by

$\boxed{ \rm{ \:y - y_1 = m(x - x_1) \: }}$

So, the equation of line $l$ which passes through the point A (a, 0) and having slope $m \: = \: - \: \dfrac{b}{a}$ is given by

$\rm \: y - 0 = - \dfrac{b}{a}(x - a)$

$\rm \: y = - \dfrac{b}{a}(x - a)$

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

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

Hence,

$\rm\implies \:\boxed{ \rm{ \:Equation \: of \: line \: is \: \dfrac{x}{a} + \dfrac{y}{b} = 1 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.