# Circle A has a center at (1 ,-8 ) and a radius of 3 . Circle B has a center at (-3 ,-2 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 4, 2016

Get the distance between the circles' centers. If the distance exceeds the sum of the radii, then the circles do not overlap. If the distance is less than the difference of the radii, the circles will also not overlap.

$D = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

$D = \sqrt{{\left(1 - - 3\right)}^{2} + {\left(- 8 - - 2\right)}^{2}}$

$\implies D = \sqrt{{4}^{2} + - {6}^{2}}$

$\implies D = \sqrt{16 + 36}$

$\implies D = \sqrt{52}$

$\implies D \approx 7.$

Since the distance between the two centers is greater than the sum of the circle's centers, the circles will not overlap.

How do we get smallest distance between the circles?
We subtract the radius of the 2 circles from the distance

$D ' = D - {r}_{1} - {r}_{2}$

$D ' \approx 7 - 3 - 2$

$\implies D ' \approx 2$

If you need the precise value, go get a calculator and compute the value of $\sqrt{52}$.