# Circle A has a radius of 4  and a center at (8 ,2 ). Circle B has a radius of 3  and a center at (4 ,5 ). If circle B is translated by <-3 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 31, 2018

$\text{no overlap } \approx 2.9$

#### Explanation:

$\text{What we have to do here is to compare the distance (d)}$
$\text{to the sum of the radii}$

• " if sum of radii">d" then circles overlap"

• " if sum of radii"< d" then no overlap"

$\text{Before calculating d we require to find the new centre}$
$\text{of B under the given translation}$

$\text{under a translation } < - 3 , 4 >$

$\left(4 , 5\right) \to \left(4 - 3 , 5 + 4\right) \to \left(1 , 9\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

$\text{to calculate d use the "color(blue)"distance formula}$

•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

$\text{let "(x_1,y_1)=(8,2)" and } \left({x}_{2} , {y}_{2}\right) = \left(1 , 9\right)$

$d = \sqrt{{\left(1 - 8\right)}^{2} + {\left(9 - 2\right)}^{2}} = \sqrt{49 + 49} = \sqrt{98} \approx 9.9$

$\text{sum of radii } = 4 + 3 = 7$

$\text{since sum of radii"< d" then no overlap}$

$\text{minimum distance "=d-" sum of radii}$

$\textcolor{w h i t e}{\times \times \times \times \times \times x} = 9.9 - 7 = 2.9$
graph{((x-8)^2+(y-2)^2-16)((x-1)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}