The geometric mean of the minimum and maximum 

values of the distance of point $(- 7, 2)$, from the points 

on the circle $x^2+y^2-10x-14y-51=0$ is equal to $m\sqrt{n}$,

 where $m, n$ are coprimes, then $mn$=

 

Let $C$be the centre of the given circle and $r$be its

radius. The coordinates of $C$are $(5, 7)$and $r=5\sqrt{5}$.

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, $CP=\sqrt{(5+7{)}^2+(7-2{)}^2}=13$

Geometric mean $=\sqrt{(13+5\sqrt{5})(13-5\sqrt{5})}=\sqrt{169-125}=2\sqrt{11}$

$\therefore$          $m = 2$ and $n = 11$. Hence, $mn$= 22