Question

If a circle and rectangular hyperbola $xy=c^2$ meets in 4 points $P,Q,RandS$ then $O{P}^2+O{Q}^2+O{R}^2+O{S}^2=$______ where r is the radius of the circle. O is the origin.

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Let the circle be $x^2+y^2+2gx+2gy+d=0$ and equation of hyperbola is $xy=c^2$

$y=\frac{c^2}{x}$

substituting y in the equation of circle

we get $x^2+\frac{c^4}{x^2}+2gx+2f\frac{c^2}{x}+d=0$

$x^2+c^4+2g{x}^3+2f{c}^{2}x+c^4=0$

$x^4+2g{x}^3+d{x}^2+2f{c}^{2}x+c^4=0$ - - - - - - (1)

This is a biquadratic equation .If should HAVE four roots.

we need to find the value of $O{P}^2+O{Q}^2+O{R}^2+O{S}^2$

Let take the co-ordinates of P$(x_1,y_1),Q(x_2,y_2)$- - - - - - -

$O{P}^2=x_1^2+y_1^2$

$O{Q}^2=x_2^2+y_2^2$

$O{R}^2=x_3^2+y_3^2$

$O{S}^2=x_4^2+y_4^2$

$O{P}^2+O{Q}^2+O{R}^2+O{S}^2=(x_1^2+x_2^2+x_3^2+x_4^2)+(y_1^2+y_2^2+y_3^2+y_4^2)$

=$\sum x_1^2+\sum y_1^2$

=$(x_1+x_2+x_3+x_4{)}^2-2\sum x_1{x}_2+\sum y_1^2$.