Question

If $P(x,y)$ is any point on the line joining the points $A(a,0)$ and $B(0,b)$ then show that $\frac{x}{a}+\frac{y}{b}=1$

Answer 1

If a point $(x,y)$ lies on a line joining the points $A(x_1,y_1)$ and $B(x_2,y_2),$ the equation of the line is given by

$\Rightarrow$ $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$

Point $P(x,y)$ lies on the line joining the points $A(a,0)$ and $B(0,b).$ So,

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

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

$\Rightarrow$ $ay=-b(x-a)$

$\Rightarrow$ $ay=-bx+ab$

$\Rightarrow$ $ay+bx=ab$

Divide both sides by $ab$

$\Rightarrow$ $\frac{ay}{ab}+\frac{bx}{ab}=\frac{ab}{ab}$

$\Rightarrow$ $\frac{y}{b}+\frac{x}{a}=1$ ----- Hence proved