The geometric mean of the minimum and maximum values of the distance of point  (- 7, 2) , from the points on the circle x2+y2−10x−14y−51=0 is equal to mn, where m, n are coprimes, then mn =

Let C be the centre of the given circle and r be itsradius. The coordinates of C are  (5, 7) and r=55.The maximum and minimum values of the distances of point P(−7,2)  from the points on the given circle are (CP+r) and(CP - r) respectively.We have, CP=(5+7)2+(7−2)2=13Geometric mean =(13+55)(13−55)=169−125=211∴          m = 2 and n = 11. Hence, mn = 22