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Question

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


A
X1+X2+X3+X4=0
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B
Y1+Y2+Y3+Y4=0
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C
X1X2X3X4=c4
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D
Y1Y2Y3Y4=c4
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Solution

The correct options are
A $$X_1\, +\, X_2\, +\, X_3\, +\, X_4\, =\, 0$$
B $$Y_1\, +\, Y_2\, +\, Y_3\, +\, Y_4\, =\,0$$
C $$Y_1\,Y_2\,Y_3\,Y_4\, =\, c^4$$
D $$X_1\, X_2\, X_3\, X_4\, =\, c^4$$
Since, the circle $$x^2\, +\, y^2\, =\, a^2$$ intersects the hyperbola $$xy\, =\, c^2$$
Therefore, $$x^2+\dfrac{c^4}{x^2}=a^2$$
$$\Rightarrow x^4-a^2x^2+c^4=0$$
now sum of the roots: $$x_1+x_2+x_3+x_4=0$$
and product of the roots $$x_1x_2x_3x_4=c^4$$
Repeat the same for $$y$$

Maths

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