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Question

# Assertion :The center of the mean position of the four points bisects the distance between the centre of the two curves. Reason: x1+x2+x3+x44=g2,y1+y2+y3+y44=f2,x1.x2.x3.x4=y1.y2.y3.y4=1.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionLet the rectangular hyperbola be xy=c2 and the equation of the circle be x2+y2+2gx+2fy+k=0.Any point on the hyperbola is (cp,cp). If it lies on the circle, then c2p2+c2p2+2gcp+2fcp+k=0. c2p4+2gcp3+kp2+2fcp+c2=0.This is fourth-degree equation in p, which has four roots. Hence the circle and the hyperbola intersect in four points.If p1,p2,p3,p4 are the roots of this equation, thenp1+p2+p3+p4=2gcc2=2gc cp1+cp2+cp3+cp4=2g x1+x2+x3+x44=g2Also 1p1+1p2+1p3+1p4=p1.p2.p3p1.p2.p3.p4=2fcc2=2fc cp1+cp2+cp3+cp4=2f y1+y2+y3+y44=f2.Hence the mean of the four points is (g2,f2) which is the mid-point of the center of the hyperbola and that of the circle. Hence, the correct option is (a).

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