Question

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

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

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

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

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

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

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

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

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

Divide both sides by $$ab$$

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

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