Question

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

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

Answer 1

The correct option is D $3x+2y=xy$

let the equation of line be $y=mx+c..........\left(1\right)$

since$(2,3)$ lies on the line,so $c=3-2m$

equation$\left(1\right)$will become $y=mx+3-2m$

$-mx+y=3-2m$

$\frac{x}{\frac{2m-3}{m}}+\frac{y}{3-2m}=1$

P and Q will be $(\frac{2m-3}{m},0)and(0,3-2m)$ respectively

since OPRQ is a rectangle$⟹$ coordinates of R will be$(\frac{2m-3}{m},3-2m)$

i.e. $x=\frac{2m-3}{m}......\left(2\right),y=3-2m.......\left(3\right)$

$\frac{\left(3\right)}{\left(2\right)}⟹\frac{y}{x}=-m.........\left(4\right)$

substitute $\left(4\right)$ in $\left(3\right)$ $⟹$ $y=3+2\frac{y}{x}$

the required locus is $3x+2y=xy$