Question

$PQ$ and $RS$ are two perpendicular chords of the rectangular hyperbola $xy = c^2.$ If C is the centre of the rectangular hyperbola, then the product of the slopes of $CP, CQ, CR$ and $CS$ is equal to

• A. -1
• B. 1
• C. 0
• D. none of these.

Answer 1

Answer: (B)

Let $t_{1}, t_{2}, t_{3}, t_{4}$ be the parameters of the pts. $P,Q,R,S$ respectively
$\therefore P$ is $ \left(ct_{1}, \frac{c}{t_{1}}\right), Q$ is $\left(ct_{2}, \frac{c}{t_{2}}\right) $
$ R$ is $\left(ct_{3}, \frac{c}{t_{3}}\right), S$ is $\left(ct_{4}, \frac{c}{t_{4}}\right)$.
$PQ\bot RS $
$ \Rightarrow \frac{\frac{c}{t_{2}}-\frac{c}{t_{1}}}{ct_{2} -ct_{1}} \cdot\frac{ \frac{c}{t_{4}}-\frac{c}{t_{3}}}{ct_{4}-ct_{3}} = -1$
$\Rightarrow - \frac{1}{t_{1} t_{2}} - \frac{1}{t_{3}t_{4}} = -1 $
$\Rightarrow t_{1}t_{2}t_{3}t_{4} = -1$
Product of slopes of $CP,CQ,CR,CS$
$ =\frac{ c /t_{1} }{c_{1}}\cdot\frac{c /t_{2}}{ct_{2}}\cdot\frac{c /t_{3}}{ct_{3}}\cdot \frac{c /t_{4}}{ct_{4}}$
$ = \frac{1}{t_{1}^{2}t_{2}^{2}t_{3}^{2}t_{4}^{2}} = \frac{1}{\left(-1\right)^{2}} = 1$