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 $C{P}^2+C{Q}^2+C{R}^2+C{S}^2=$

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

Answer 1

The correct option is D $4r^2$

Let $xy=c^2$ is rectangular hyperbola and $x^2+y^2=a^2$ is the circle and having centre $O(0,0)$
Let $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ are coordinates of $A,B,C,D$
Since $A,B,C,D$ lie on the circle and the hyperbola so,
'$x_1^2+y_1^2=r^2,x_2^2+y_2^2=r^2,x_3^2+y_3^2=r^2,x_4^2+y_4^2=r^2,$
Therefore $O{A}^2+O{B}^2+O{C}^2+O{D}^2=(x_1^2+y_1^2)+(x_2^2+y_2^2)+(x_3^2+y_3^2)+(x_4^2+y_4^2)=4r^2$