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 :

Straight Lines and Pair of Straight Lines PYQ — JEE Main 2018 · Shift 0 · Q37 with Solution

View written solutionFree

Correct answer: D

Let the line cut the $x$ -axis at $P(a,0)$ and the $y$ -axis at $Q(0,b)$ , where $a\neq 0$ , $b\neq 0$ .
Since the rectangle $OPRQ$ is completed with $O=(0,0)$ , $P=(a,0)$ , $Q=(0,b)$ , the fourth vertex is $R=(a,b).$ So if $R=(x,y)$ , then $a=x,\quad b=y.$
The line joining $P(a,0)$ and $Q(0,b)$ has intercept form $\frac{X}{a}+\frac{Y}{b}=1.$ It passes through the fixed point $(2,3)$ , so $\frac{2}{a}+\frac{3}{b}=1.$
Now substitute $a=x$ and $b=y$ : $\frac{2}{x}+\frac{3}{y}=1.$ Multiplying by $xy$ , $2y+3x=xy.$ Rearranging, $3x+2y=xy.$
Therefore, the locus of $R$ is $\boxed{3x+2y=xy}.$
Checking options:

Hence the correct option is D.