If the circle x^2 + y^2 = a^2 intersects the | vedclass

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(D) Given the equations $x^2 + y^2 = a^2$ and $xy = c^2$.Substituting $y = \frac{c^2}{x}$ into the circle equation:$x^2 + \frac{c^4}{x^2} = a^2$$x^4 -

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All of the above

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(D) Given the equations $x^2 + y^2 = a^2$ and $xy = c^2$.Substituting $y = \frac{c^2}{x}$ into the circle equation:$x^2 + \frac{c^4}{x^2} = a^2$$x^4 - a^2 x^2 + c^4 = 0$This is a biquadratic equation in $x$. The roots are $x_1, x_2, x_3, x_4$.From the properties of roots,the sum of roots $x_1 + x_2 + x_3 + x_4 = 0$ (coefficient of $x^3$ is $0$) and the product of roots $x_1 x_2 x_3 x_4 = c^4$.Since the equations are symmetric with respect to $x$ and $y$,by symmetry,$y_1 + y_2 + y_3 + y_4 = 0$ and $y_1 y_2 y_3 y_4 = c^4$.Thus,all the given options are correct.

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

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