# Circle A has a center at (1 ,-2 ) and a radius of 3 . Circle B has a center at (-2 ,6 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Apr 24, 2018

$\textcolor{b l u e}{\sqrt{73} - 5}$

#### Explanation:

Let $d = \text{the distance between centres}$

Let $s = \text{the sum of the radii}$

Then if:

$d = s$ the circles touch at one point.

$d > s$ the circles do not touch.

$d < s$ the circles intersect at two points.

We find the distance between centres using the distance formula:

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

We have:

$A = \left(1 , - 2\right)$

$B = \left(- 2 , 6\right)$

$d = \sqrt{{\left(1 - \left(- 2\right)\right)}^{2} + {\left(\left(- 2\right) - 6\right)}^{2}} = \sqrt{73}$

Radius of $A = 3$

Radius of $B = 2$

$3 + 2 = 5$

$\therefore$

$\sqrt{73} > 5$

The circles do not touch.

The shortest distance between the circles is:

$d - s$

$\sqrt{73} - 5$

See diagram: