# Circle A has a radius of 4  and a center of (6 ,2 ). Circle B has a radius of 2  and a center of (5 ,3 ). If circle B is translated by <-2 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Apr 6, 2016

The degree of overlap is

$6 - 3 \sqrt{2} \approx 1.757$ to 3 decimal places

#### Explanation:

For circle A
Let the centre be ${C}_{a} \to \left(6 , 2\right)$
Let radius be ${R}_{a} \to 4$

For circle B
Let the centre be ${C}_{b} \to \left(5 , 3\right)$
Let the radius be ${R}_{b} \to 2$

Let distance between centres be $D$

${C}_{b}$ translated by $< - 2 , 2 >$

$\implies {C}_{b} \to \left(5 - 2 , 3 + 2\right)$

$\implies {C}_{b} \to \left(3 , 5\right)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine final distance between centres}}$

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

$D = \sqrt{{\left(3 - 6\right)}^{2} + {\left(5 - 2\right)}^{2}}$

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

$\textcolor{b l u e}{D = \sqrt{18} = 3 \sqrt{2}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine if circles overlap}}$

For this to be true we need $D < {R}_{a} + {R}_{b}$

${R}_{a} + {R}_{B} = 4 + 2 = 6$

$3 \sqrt{2} < 6$ so they do overlap

The degree of overlap is

$6 - 3 \sqrt{2} \approx 1.757$ to 3 decimal places