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 = $\frac{c^{2}}{x}$in equation (2), we get

x² + $\frac{c^{4}}{x^{2}}$ = r² Ž x⁴ – r²x² + c⁴ = 0 … (3)

Now, CP² + CQ² + CR² + CS²

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

= $\left( \sum_{i = 1}^{4}x_{i} \right)^{2}$– 2 S x₁x₂ + $\left( \sum_{i = 1}^{4}y_{i} \right)^{2}$– 2 S y₁y₂

= 2r² + 2r² = 4r². [From (3)]