Question

A rectangular hyperbola whose center is $C$, is cut by any circle of radius $r$ in four points $P,Q,R,S$. Then $C{P}^2+C{Q}^2+C{R}^2+C{S}^2=$

• A. $4r^2$
• B. $2r^2$
• C. $r^2$
• D. $8r^2$

Answer 1

The correct option is A $4r^2$

Taking the rectangular asymptotes as the axis of refernce, the equations of the hyperbola and the circle are
$xy=k^2\dots \left(i\right)x^2+y^2+2gx+2fy+c=0\dots \left(ii\right)\Rightarrow r^2=g^2+f^2-c\dots \left(iii\right)$

Eliminating $y$ from above equation
$x^4+2g{x}^3=c{x}^2+2f{k}^{2}x+k^4$

Let the roots of the equations are $x_1,x_2,x_3,x_4$
Then,
$\sum x_i=-2g\sum x_i{x}_j=c$

$\therefore \sum x_i^2={(\sum x_i)}^2-2\sum x_i{x}_j\Rightarrow \sum x_i^2=4g^2-2c$

Similarly, eliminating $x$
$\sum y_i^2=4f^2-2c$

$\therefore C{P}^2+C{Q}^2+C{R}^2+C{S}^2=x_1^2+y_1^2+\cdots =4g^2-2c+4f^2-2c=4r^2$