# Circle A has a radius of 2  and a center of (2 ,5 ). Circle B has a radius of 3  and a center of (3 ,8 ). If circle B is translated by <4 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jun 16, 2018

$\text{no overlap } , \approx 0.385$

#### Explanation:

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

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

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

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

$\text{under the translation } < 4 , - 1 >$

$\left(3 , 8\right) \to \left(3 + 4 , 8 - 1\right) \to \left(7 , 7\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(2,5)" and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 7\right)$

$d = \sqrt{{\left(7 - 2\right)}^{2} + {\left(7 - 5\right)}^{2}} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385$

$\text{sum of radii } = 2 + 3 = 5$

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

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

$\textcolor{w h i t e}{\times \times \times \times \times \times x} = 5.385 - 5 = 0.385$
graph{((x-2)^2+(y-5)^2-4)((x-7)^2+(y-7)^2-9)=0 [-20, 20, -10, 10]}