Question

A rectangular hyperbola whose cente is C is cut by any circle of radius r in four point P, Q, R, S. The value of $C{P}^2+C{Q}^2+C{R}^2+C{S}^2$ is equal to :

• A. $r^2$
• B. $2r^2$
• C. $3r^2$
• D. $4r^2$

Answer 1

The correct option is D $4r^2$

Let the equation of the rectangular hyperbola be $xy=c^2$ ...(1)

And, equation of circle be $x^2+y^2=r^2$ ....(2)

Put $y=\frac{c^2}{x}$ in equation 2, we get,

$x^2+\frac{c^4}{x^2}=r^2$

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

Now, $C{P}^2+C{Q}^2+C{R}^2+C{S}^2$

$={x_1}^2+{y_1}^2+{x_2}^2+{y_2}^2+{x_3}^2+{y_3}^2+{x_4}^2+{y_4}^2$

$=2r^2+2r^2$

$=4r^2$