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

Feb 8, 2018

$\text{circles overlap}$

#### Explanation:

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

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

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

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

$\left(4 , 7\right) \to \left(4 + 2 , 7 - 4\right) \to \left(6 , 3\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(2,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 3\right)$

$d = \sqrt{{\left(6 - 2\right)}^{2} + {\left(3 - 4\right)}^{2}} = \sqrt{16 + 1} = \sqrt{17} \approx 4.123$

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

$\text{since sum of radii">d" then circles overlap}$
graph{((x-2)^2+(y-4)^2-9)((x-6)^2+(y-3)^2-4)=0 [-20, 20, -10, 10]}