Question

If 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. $y_1 + y_2 + y_3 + y_4 = 2$
• C. $x_1 x_2 x_3 x_4 = 2 c^4$
• D. $y_1 y_2 y_3 y_4 = 2 c ^4$

Answer 1

Answer: (A)

Given, $x^{2}+y^{2}=a^{2}$ and $x y=c^{2}$
$\therefore \,\,\,\,\,\, x^{2}+\left(\frac{c^{2}}{x}\right)^{2}=a^{2} $
$\Rightarrow \,\,\,\,\,\, x^{4}-a^{2} x^{2}+c^{4}=0 $
$\therefore \,\,\,\,\,\, x_{1}+x_{2}+x_{3}+x_{4}=0$