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
B
Y1+Y2+Y3+Y4=0
C
X1X2X3X4=c4
D
Y1Y2Y3Y4=c4

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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