The correct option is
A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Let 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=g2
Also 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).