# Circle A has a radius of 1  and a center of (1 ,7 ). Circle B has a radius of 2  and a center of (8 ,1 ). If circle B is translated by <-4 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 23, 2018

$\text{no overlap } \approx 1.24$

#### Explanation:

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

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

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

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

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

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

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

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

$\text{let "(x_1,y_1)=(1,7)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 4\right)$

$d = \sqrt{{\left(4 - 1\right)}^{2} + {\left(4 - 7\right)}^{2}} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24$

$\text{sum of radii } = 1 + 2 = 3$

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

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

$\textcolor{w h i t e}{\times \times \times \times \times} = 4.24 - 3 = 1.24$
graph{((x-1)^2+(y-7)^2-1)((x-4)^2+(y-4)^2-4)=0 [-20, 20, -10, 10]}