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, $C{P}^2+C{Q}^2+C{R}^2+C{S}^2$ 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 equation of the rectangular hyperbola be $xy=c^2$ and equation of circle be $x^2+y^2=r^2$
Put $y=\frac{c^2}{x}$ in equation (ii), we get $x^2+\frac{c^4}{x^2}{r}^2$
$x^4-r^2{x}^2+c^4=0Now,C{P}^2+C{Q}^2+C{R}^2+C{S}^2$
$=x_1^2+y_1^2+x_2^2+y_2^2+x_3^2+y_3^2+x_4^2+y_4^2$
$={\left({\sum }_{i=1}^4{x}_i\right)}^2-2\sum x_1{x}_2+={\left({\sum }_{i=1}^4{y}_i\right)}^2-2\sum y_1{y}_2$
$=2r^2+2r^2=4r^2$ [from equation (iii)]