Question

If p(x,y) is any point on the line joining the points A(a, 0) and B(0, b), then show that x/a + y/b = 1.

Answer 1

If a point (x,y) lies on a line joining the points A(x₁,y₁) and B(x₂,y₂),

The equation of the line is given by  

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

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

So,

$\frac{y-0}{x-a} = \frac{b-0}{0-a} \\ \\ \Rightarrow \frac{y}{x-a}=- \frac{b}{a} \\ \\ \Rightarrow ay=-b(x-a)\\ \\ \Rightarrow ay=-bx+ab\\ \\ \Rightarrow ay+bx=ab\\ \\divide\ both\ sides\ by\ ab\\ \\ \Rightarrow \frac{ay}{ab}+ \frac{bx}{ab} = \frac{ab}{ab}\\ \\ \Rightarrow \frac{y}{b}+ \frac{x}{a} =1\ (proved)$