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)

1

Answer 2

Let $t_{1},t_{2},t_{3},t_{4}$ be the parameters of the points P, Q, R and S respectively. Then, the coordinates of P, Q, R and S are $\left( ct_{1},\frac{c}{t_{1}} \right)$, $\left( ct_{2},\frac{c}{t_{2}} \right)$, $\left( ct_{3},\frac{c}{t_{3}} \right)$ and $\left( ct_{4},\frac{c}{t_{4}} \right)$ respectively.

Now, $PQ\bot RS$ ⇒ $\frac{\frac{c}{t_{2}} - \frac{c}{t_{1}}}{ct_{2} - ct_{1}} \times \frac{\frac{c}{t_{4}} - \frac{c}{t_{3}}}{ct_{4} - ct_{3}} = - 1$

⇒ $- \frac{1}{t_{1}t_{2}} \times - \frac{1}{t_{3}t_{4}} = - 1$ ⇒ $t_{1}t_{2}t_{3}t_{4} = - 1$.....(i)

$\therefore$Product of the slopes of $CP,CQ,CR$ and $CS$

$\frac{1}{t_{1}^{2}} \times \frac{1}{t_{2}^{2}} \times \frac{1}{t_{3}^{2}} \times \frac{1}{t_{4}^{2}} = \frac{1}{t_{1}^{2}t_{2}^{2}t_{3}^{2}t_{4}^{2}} = 1$ [Using (i)]