Question

A circle cuts the rectangular hyperbola $xy=1$ in points $(x_r,y_r)$, $r=1,2,3,4$ then

• A. $x_1{x}_2{x}_3{x}_4=-1$
• B. $x_1{x}_2+x_3{x}_4=1$
• C. $x_1{x}_2{x}_3{x}_4=1$
• D. $x_1+x_2+x_3+x_4=0$

Answer 1

The correct option is C $x_1{x}_2{x}_3{x}_4=1$

Let the circle is $x^2+y^2+2gx+2fy+c=0$
So for point of intersection of circle and given hyperbola
$x^2+\frac{1}{x^2}+2gx+\frac{2f}{x}+c=0$
$x^4+2g{x}^3+c{x}^2+2fx+1=0$
Sum of roots $=x_1+x_2+x_3+x_4=-2g$
Products of roots $=x_1{x}_2{x}_3{x}_4=1$
Hence, option 'C' is correct.