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

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(A) Given equations are $x^2+y^2=a^2$ and $xy=c^2$.From the second equation,$x = \frac{c^2}{y}$.Substituting this into the first equation:$\left(\frac

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(A) Given equations are $x^2+y^2=a^2$ and $xy=c^2$.From the second equation,$x = \frac{c^2}{y}$.Substituting this into the first equation:$\left(\frac{c^2}{y}\right)^2 + y^2 = a^2$$\frac{c^4}{y^2} + y^2 = a^2$$c^4 + y^4 = a^2 y^2$$y^4 - a^2 y^2 + c^4 = 0$This is a biquadratic equation in $y$. Let the roots be $y_1, y_2, y_3, y_4$.The sum of the roots of the equation $y^4 + 0y^3 - a^2 y^2 + 0y + c^4 = 0$ is given by the coefficient of $y^3$ divided by the coefficient of $y^4$ with a negative sign.Sum $= -\frac{0}{1} = 0$.Therefore,$y_1+y_2+y_3+y_4 = 0$.

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

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