Question

A straight line through a fixed point $(2,3)$ intersects the coordinates 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

Answer: (A)

$3x+2y=xy$

$y-y_1=m(x-x_1)$

$\therefore$ $y-3=m(x-2)$

$\Rightarrow mx-y-2m+3=0$

$\Rightarrow mx-y=2m-3$

X-intercept $=\frac{2m-3}{m}$

Y-intercept $=3-2$m

$\therefore$ Co-ordinates of rectangle $(0,0)$

$\Rightarrow \left(\frac{2m-3}{m},0\right)$

$\Rightarrow (0,3-2m)$

$\therefore$ Co-ordinates of R is $\left(\frac{2m-3}{m},3-2m\right)$

$\Rightarrow x=\frac{2m-3}{m}$ & $y=3-2m$

$y=3-2m$

$\Rightarrow m=\frac{3-y}{2}$

$\therefore x=\frac{2m-3}{m}$

$=2-\frac{3}{m}$

$x=2-\frac{3\left(2\right)}{(3-y)}$

$\Rightarrow x(3-y)=2(3-4)-6$

$\Rightarrow 3x-xy=6-2y-6$

$\Rightarrow 3x+2y=xy$ is the locus.

options (1)