Question

If the circle $x^2+y^2=1$ cuts the rectangular hyperbola $xy=1$ in four points $(x_i,y_i)i=1,2,3,4$ then

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

Answer 1

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

Let $(x_i,y_i)=(t,\frac{1}{t})$ for $I=1,2,3,4$

As these points lie on the circle, $t^2+\frac{1}{t^2}=1\Rightarrow t^4-t^2+1=0$

$\Rightarrow t_1\times t_2\times t_3\times t_4=1$

$\frac{1}{t_1{t}_2{t}_3{t}_4}=1\Rightarrow y_1{y}_2{y}_3{y}_4=1$