Question

A circle cuts the rectangular hyperbola $xy=1$ in points $(x_r,y_r),r=1,2,3,4$. If $x_1{x}_2{x}_3{x}_4=a,y_1{y}_2{y}_3{y}_4=b$, then $a^2+b^2=$

Answer 1

Let the equation of the circle be
$x^2+y^2+2gx+2fy+c=0\cdots \left(1\right)$

and the equation of the given hyperbola is
$xy=1\cdots \left(2\right)$

On solving, we get
$x^2+\frac{1}{x^2}+2gx+\frac{2f}{x}+x=0$
$\Rightarrow x^4+2g{x}^3+c{x}^2+2fx+1=0$

Let its roots are $x_1,x_2,x_3$ and $x_4$
Then, $x_1{x}_2{x}_3{x}_4=1$
Similarly, $y_1{y}_2{y}_3{y}_4=1$
$\therefore a^2+b^2=2$