Straight Line and Pair of Straight Lines 1 Question 18
18. 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 $O P R Q$ is completed, then the locus of $R$ is
(2018 Main)
(a) $3 x+2 y=6$
(b) $2 x+3 y=x y$
(c) $3 x+2 y=x y$
(d) $3 x+2 y=6 x y$
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Solution:
Equation of line $P Q$ is $\frac{x}{\alpha}+\frac{y}{\beta}=1$
Since this line is passes through fixed point $(2,3)$.
$$ \therefore \quad \frac{2}{\alpha}+\frac{3}{\beta}=1 $$
$\therefore$ Locus of $R$ is
$$ 2 \beta+3 \alpha=\alpha \beta $$
i.e.
$$ 2 y+3 x=x y \Rightarrow 3 x+2 y=x y $$
