The geometric mean of the minimum and maximum
values of the distance of point $(- 7, 2)$ , from the points
on the circle $x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ is equal to $m \sqrt{n}$ ,
where $m, n$ are coprimes, then $m n$ =

Question

The geometric mean of the minimum and maximum

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

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

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

Infinity Learn · Accepted Answer

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 $$=$$ $$22$$

Definition of a circle

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The geometric mean of the minimum and maximum
values of the distance of point $(- 7, 2)$ , from the points
on the circle $x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ is equal to $m \sqrt{n}$ ,
where $m, n$ are coprimes, then $m n$ =

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Definition of a circle

Remember concepts with our Masterclasses.

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 =

Ready to Test Your Skills?

free study material

The geometric mean of the minimum and maximum
values of the distance of point $(- 7, 2)$ , from the points
on the circle $x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ is equal to $m \sqrt{n}$ ,
where $m, n$ are coprimes, then $m n$ =