Question

A line is drawn through a fixed point $ (h,k) $ cutting the coordinate axes at P and Q respectively. The rectangle OPRQ is completed. Find the equation of locus of R.

• A. $ \frac{x}{h}+\frac{y}{k}=1 $
• B. $ \frac{x}{y}+\frac{h}{k}=1 $
• C. $ \frac{h}{x}+\frac{k}{y}=1 $
• D. $ \frac{x}{k}+\frac{y}{h}=1 $

Answer 1

Answer: (C)

Let the coordinates of $R(\alpha, \beta)$. Since, OPRQ is a rectangle, therefore coordinates of $P$ and $Q$ are $(\alpha, 0)$ and $(0, \beta)$ respectively. Now, equation of line $P Q$ is
$\frac{x}{\alpha}+\frac{y}{\beta}=1$
Since, $(h, k)$ lies on this line
$\therefore \frac{h}{\alpha}+\frac{k}{\beta}=1$

Hence, locus of $R(\alpha, \beta)$ is
$\frac{h}{x}+\frac{k}{y}=1$