Question

A rectangular hyperbola whose centre is C is cut by any circle of radius r in four points P, Q, R and S. Then

CP² + CQ² + CR² + CS² is equal to

• A. r²
• B. 2r²
• C. 3r²
• D. 4r²

Answer 1

Answer: (D)

4r²

Answer 2

Let equation of the rectangular hyperbola be

xy = c² ... (1)

and equation of circle be x² + y² = r² ... (2)

Put y = c²/x in (2), we get

$x^{2} + \frac{c^{4}}{x^{2}} = r^{2} \Rightarrow x^{4} - r^{2}x^{2} + c^{4} = 0$... (3)

Now, CP² + CQ² + CR² + CS²

$= {x_{1}}^{2} + {y_{1}}^{2} + {x_{2}}^{2} + {y_{2}}^{2} + {x_{3}}^{2} + {y_{3}}^{2} + x_{4}^{2} + {y_{4}}^{2}$.

= $\left( \sum_{i = 1}^{4}{xi} \right)^{2} - 2\Sigma x_{1}x_{2} + \left( \sum_{i = 1}^{4}{yi} \right)^{2} - 2\Sigma y_{1}y_{2}$.

= 2r² + 2r² = 4r².