Let S1:x2+y2=9 and S2:(x-2)2+y2=1 . Then the locus of center of a variable  circle S which touches S1 internally and S2 externally always passes through the points:

Suppose that the center of the circle S3 be h,kThe sqaure of the distance from origin to the point (h,k) is r-32The sqaure of the distance from  2,0 to the point h,k is r+22h2+k2=(r-3)2=r2-6r+9 (h-2)2+k2=(r+1)2=r2+2r+14h-4=-8r+8⇒r=3-h2 h2+k2=(3+h)24 Locus is 3(x-1)2+4y2=12 ∴2,±32 lies on the locus.