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

Answer 1

The correct option is A

$3x+2y=xy$

Explanation for the correct answer:

Finding the locus of $R$

The general equation of straight line through point is

$\frac{x}{h}+\frac{y}{k}=1$

where $h$ is the $x-$intercept and $k$ is the $y-$intercept.

The line passes through $(2,3)$, so we get

$$\begin{gathered} \Rightarrow \frac{2}{h}+\frac{3}{k}=1 \\ 2k+3h=hk \end{gathered}$$

Now substitute $h$ as $x$ and $k$ as $y$, we get the locus of $R$ as

$\Rightarrow 3x+2y=xy$

Hence, option (A) $\mathbf{3}\mathit{x}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathit{x}\mathit{y}$ is the correct answer.