Question

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

• A. $x_1{x}_2{x}_3{x}_4=2$
• B. $x_1{x}_2{x}_3{x}_4=1$
• C. $x_1+x_2+x_3+x_4=0$
• D. $y_1+y_2+y_3+y_4=0$

Answer 1

Answer: (B)

Let the circle be
$x^2+y^2+2gx+2fy+K=0...\left(1\right)$
and hyperbola be $xy=1...\left(2\right)$
From $\left(2\right),y=\frac{1}{x}$. Putting in $\left(1\right)$, we get
$x^2+\frac{1}{x^2}+2gx+\frac{2f}{x}+K=0$
i.e., $x^4+2g{x}^3+K{x}^2+2fx+1=0$
Let its root be $x_1,x_2,x_3,x_4$
$\therefore x_1{x}_2{x}_3{x}_4=1$