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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=c4
xy=c2
x=c2y
Substituting the value of x in the equation of the circle, we get,
c4y2+y2=a2
y4a2y2+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,
x4a2x2+c4=0
x1+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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