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 $\left(c{t}_1,\frac{c}{t_1}\right),Q$ is $\left(c{t}_2,\frac{c}{t_2}\right)$
$R$ is $\left(c{t}_3,\frac{c}{t_3}\right),S$ is $\left(c{t}_4,\frac{c}{t_4}\right)$.
$PQ\perp RS$
$\Rightarrow \frac{\frac{c}{t_2}-\frac{c}{t_1}}{c{t}_2-c{t}_1}\cdot \frac{\frac{c}{t_4}-\frac{c}{t_3}}{c{t}_4-c{t}_3}=-1$
$\Rightarrow -\frac{1}{t_1{t}_2}-\frac{1}{t_3{t}_4}=-1$
$\Rightarrow t_1{t}_2{t}_3{t}_4=-1$
Product of slopes of $CP,CQ,CR,CS$
$=\frac{c/t_1}{c_1}\cdot \frac{c/t_2}{c{t}_2}\cdot \frac{c/t_3}{c{t}_3}\cdot \frac{c/t_4}{c{t}_4}$
$=\frac{1}{t_1^2{t}_2^2{t}_3^2{t}_4^2}=\frac{1}{{\left(-1\right)}^2}=1$