Circle A has a center at (2 ,4 ) and an area of 81 pi. Circle B has a center at (4 ,3 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Nov 7, 2016

The larger circle completely encloses the smaller circle.

Explanation:

Here is a graph of the two circles:

Nov 8, 2016

Smaller circle lies within bigger circle. Smallest distance between them is $0.764$

Explanation:

Circle A has a center at $\left(2 , 4\right)$ and its radius is $\sqrt{\frac{81 \pi}{\pi}} = 9$

Circle B has a center at $\left(4 , 3\right)$ and its radius is $\sqrt{\frac{36 \pi}{\pi}} = 6$

The distance between centers is

$\sqrt{{\left(4 - 2\right)}^{2} + {\left(3 - 4\right)}^{2}} = \sqrt{{2}^{2} + {1}^{2}} = \sqrt{5}$

As the distance at $\sqrt{5}$ between them is less than difference in radius which is $9 - 6 = 3$

smaller circle lies within bigger circle. For details see here

Also see graph below (the other answer).

Smallest distance between them is $9 - 6 - \sqrt{5} = 0.764$