Question

A straight line through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:

• A. 3x + 2y = 6
• B. 2x + 3y = xy
• C. 3x + 2y = xy
• D. 3x + 2y = 6xy

Answer 1

Answer: (C)

3x + 2y = xy

Answer 2

Let the equation of line be $\frac{x}{a}+\frac{y}{b}=1$
(i) passes through the fixed point $(2,3)$
$\Rightarrow \frac{2}{a}+\frac{3}{b}=1$
$P(a, 0), Q(0, b), O(0,0)$, Let $R(h, k)$

Midpoint of $O R$ is $\left(\frac{h}{2}, \frac{k}{2}\right)$
Midpoint of $P Q$ is $\left(\frac{a}{2}, \frac{b}{2}\right)$
$\Rightarrow h=a, \,\,\,\,\, k=b \,\,\,\,\, \ldots$ (iii)
From (ii) & (iii),
$\frac{2}{h}+\frac{3}{k}=1 \,\,\,\,\, \Rightarrow$ locus of $R(h, k)$
$\frac{2}{x}+\frac{3}{y}=1 \,\,\,\,\, \Rightarrow 3 x+2 y=x y$