Question

PQ and RS are two perpendicular chords of the rectangular hyperbola xy = c². 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, t₂, t₃ and t₄ be the parameters of the point 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 is perpendicular to RS

Ž $\frac{\frac{c}{t_{2}} - \frac{c}{t_{1}}}{ct_{2} - ct_{1}}$ × $\frac{\frac{c}{t_{4}} - \frac{c}{t_{3}}}{ct_{4} - ct_{3}}$= – 1 Ž $\frac{1}{t_{1}t_{2}}$ × – $\frac{1}{t_{3}t_{4}}$ = – 1

Ž t₁ t₂ t₃ t₄ = – 1 … (1)

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

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