Question

lf the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2)$, $R(x_3,y_3)$ and $S(x_4,y_4)$ , 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)

The correct options are
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 $y_1{y}_2{y}_3{y}_4=c^4$

$xy=c^2$
$\Rightarrow x=\frac{c^2}{y}$
Substituting the value of $x$ in the equation of the circle, we get,
$\frac{c^4}{y^2}+y^2=a^2$
$\Rightarrow y^4-a^2{y}^2+c^4=0$
$\Rightarrow y_1+y_2+y_3+y_4=0$ (sum of roots)
$y_1{y}_2{y}_3{y}_4=c^4$ (product of roots)
The equations of the circle and the hyperbola are symmetric in $x$ and $y$. Hence,
$x^4-a^2{x}^2+c^4=0$
$x_1+x_2+x_3+x_4=0$ (sum of roots)
$x_1{x}_2{x}_3{x}_4=c^4$ (product of roots)
Hence, options $A,B,C$ and $D$ are all correct.