Question

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ at four points
$P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),andS(x_4,y_4)$, then

• A. $x_1+x_2+x_3+x_4=0$
• B. $y_1+y_2+y_3+y_4=0$
• C. $x_1{x}_2{x}_3{x}_4=c^4$
• D. $y_1{y}_2{y}_3{y}_4=c^4$

Answer 1

Answer: (A), (B), (C), (D)

$y_1{y}_2{y}_3{y}_4=c^4$

Putting $y=\frac{c^2}{x}in{x}^2+y^2=a^2$, we get
$x^2+\frac{c^4}{x^2}=a^{2}or{x}^4-a^2{x}^2+c^4=0As{x}_1,x_2,x_3,and{x}_4$ are the roots of (i), we have
$x_1+x_2+x_3+x_4=0and{x}_1{x}_2{x}_3{x}_4=c^4$
Similarly, forming equation in y, we get
$y_1+y_2+y_3+y_4=0and{y}_1{y}_2{y}_3{y}_4=c^4$