If the circle $x^{2} + y^{2} = a^{2}$ intersects the hyperbola xy = $c^{2}$ in four points $(x_{i}, y_{i}),$ foci = 1, 2, 3 and 4, then $y_{1} + y_{2} + y_{3} + y_{4}$ equals

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c^{4}

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Solution

The correct option is A 0
Given,
$x^{2} y^{2} = c^{4}$
$\Rightarrow y^{2} (a^{2} - y^{2}) = c^{4}$
$\Rightarrow y^{4} - a^{2} y^{2} + c^{4} = 0$
Let $y_{1} + y_{2} + y_{3}$ and $y_{4}$ are the roots.
$∴ y_{1} + y_{2} + y_{3} + y_{4} = 0$

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x y = c^{2}

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If the circle x2+y2=a2 intersects the hyperbola xy = c2 in four points (xi,yi), foci = 1, 2, 3 and 4, then y1+y2+y3+y4 equals

The correct option is A 0Given,x2y2=c4⇒y2(a2−y2)=c4⇒y4−a2y2+c4=0Let y1+y2+y3 and y4 are the roots.∴y1+y2+y3+y4=0

If the circle $x^{2} + y^{2} = a^{2}$ intersects the hyperbola xy = $c^{2}$ in four points $(x_{i}, y_{i}),$ foci = 1, 2, 3 and 4, then $y_{1} + y_{2} + y_{3} + y_{4}$ equals

The correct option is A 0
Given,
$x^{2} y^{2} = c^{4}$
$\Rightarrow y^{2} (a^{2} - y^{2}) = c^{4}$
$\Rightarrow y^{4} - a^{2} y^{2} + c^{4} = 0$
Let $y_{1} + y_{2} + y_{3}$ and $y_{4}$ are the roots.
$∴ y_{1} + y_{2} + y_{3} + y_{4} = 0$