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

Option: 1

-1


Option: 2

1


Option: 3

0


Option: 4

None of these


Answers (1)

Let \mathrm{t_1, t_2, t_3, t_4} be the parameters of the points\mathrm{P,Q,R} and \mathrm{S} respectively. Then, the coordinates of \mathrm{P, Q, R} and \mathrm{S} are
\mathrm{\left(c t_1, \frac{c}{t_1}\right) \quad\left(c t_2, \frac{c}{t_2}\right) \quad\left(c t_3, \frac{c}{t_3}\right) \text { and }\left(c t_4, \frac{c}{t_4}\right)} respectively.
Now, \mathrm{P Q \perp R S \Rightarrow \frac{\frac{c}{t_2}-\frac{c}{t_1}}{c t_2-c t_1} \times \frac{\frac{c}{t_4}-\frac{c}{t_3}}{c t_4-c t_3}=-1 \Rightarrow-\frac{1}{t_1 t_2} \times-\frac{1}{t_3 t_4}=-1 \Rightarrow t_1 t_2 t_3 t_4=-1\ \ ..........(i)}
\therefore Product of the slopes of \mathrm{C P, C Q, C R \text { and } C S}
\mathrm{\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\ \ \ \\ \ \ \left [ Using (i) \right ]}

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Ramraj Saini

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