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Parametric Equation of Parabola

PQ, RS are tw...

Question

PQ, RS are two perpendicular chords of the hyperbola $x y = c^{2} .$ If T is the centre of hyperbola, then product of the slopes of TP, TQ, TR, TS is

A

-1

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B

1

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C

tending to infinity

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D

0.0

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Solution

The correct option is B 1
Let coordinates of P, Q, R, S be respectively.
$(c t_{1}, \frac{c}{t_{1}}), (c t_{2}, \frac{c}{t_{2}}), (c t_{3}, \frac{c}{t_{3}}), (c t_{4}, \frac{c}{t_{4}}) P Q ⊥ R S \Rightarrow (\frac{\frac{c}{t_{2}} - \frac{c}{t_{1}}}{c t_{2} - c t_{1}}) (\frac{\frac{c}{t_{4}} - \frac{c}{t_{3}}}{c t_{4} - c t_{3}}) = - 1 \Rightarrow (- \frac{1}{t_{1} t_{2}}) (- \frac{1}{t_{3} t_{4}}) = - 1 \Rightarrow t_{1} t_{2} t_{3} t_{4} = - 1$
Now product of slopes i.e., $\frac{1}{t_{1}^{2}} . \frac{1}{t_{2}^{2}} . \frac{1}{t_{3}^{2}} . \frac{1}{t_{4}^{2}} = (- 1)^{2} = 1$

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