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Question

# lf the circle x2+y2=a2 intersects the hyperbola xy=c2 in four points P(x1,y1),Q(x2,y2), R(x3,y3) and S(x4,y4) , 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 x1+x2+x3+x4=0 B y1+y2+y3+y4=0 C x1x2x3x4=c4 D y1y2y3y4=c4xy=c2⇒x=c2ySubstituting the value of x in the equation of the circle, we get, c4y2+y2=a2⇒y4−a2y2+c4=0⇒y1+y2+y3+y4=0 (sum of roots)y1y2y3y4=c4 (product of roots)The equations of the circle and the hyperbola are symmetric in x and y. Hence,x4−a2x2+c4=0x1+x2+x3+x4=0 (sum of roots)x1x2x3x4=c4 (product of roots)Hence, options A,B,C and D are all correct.

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