If a circle and the rectangular hyperbola xy=c2 meet in the four points p1=1,p2=2,p3=3 and p4, and the center of the circle through the points p1,p2 and p3 is (3712,4712) (where p is the parameter of the parametric coordinates of the hyperbola), find c.
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, then
p1+p2+p3+p4=2gcc2=2gc
cp1+cp2+cp3+cp4=−2g
x1+x2+x3+x44=g/2 ..(i)
Also 1p1+1p2+1p3+1p4=p1.p2.p3p1.p2.p3.p4=2fcc2=2fc
cp1+cp2+cp3+cp4=2f
y1+y2+y3+y44=f2 ..(ii)
Also, p1.p2.p3.p4=c2c2=1 ..(iii)
Hence from (i),(ii) and (iii) we can clearly see that
−g,−f=c2(p1+p2+p3+1p1.p2.p3),c2(1p1+1p2+1p3+p1.p2.p3)
Hence, comparing the centers, we get c=1.