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

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

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$x_{1}+x_{2}+x_{3}+x_{4}=0$

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(D) Given equations are $x^{2}+y^{2}=a^{2}$ and $xy=c^{2}$.Substituting $y = \frac{c^{2}}{x}$ into the circle equation:$x^{2} + (\frac{c^{2}}{x})^{2} = a^{2}$$x^{2} + \frac{c^{4}}{x^{2}} = a^{2}$$x^{4} - a^{2}x^{2} + c^{4} = 0$This is a biquadratic equation in $x$. The roots are $x_{1}, x_{2}, x_{3}, x_{4}$.Comparing this with the standard form $Ax^{4} + Bx^{3} + Cx^{2} + Dx + E = 0$,we have $B=0$.By Vieta's formulas,the sum of the roots $x_{1}+x_{2}+x_{3}+x_{4} = -\frac{B}{A} = 0$.

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

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