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.

$$ 2 y+3 x=x y \Rightarrow 3 x+2 y=x y $$



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