Circle A has a radius of 5  and a center of (2 ,7 ). Circle B has a radius of 4  and a center of (7 ,3 ). If circle B is translated by <-1 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jan 21, 2018

$\text{circles overlap}$

Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance (d) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of the radii}}$

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

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

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

$\text{under a translation } < - 1 , 2 >$

$\left(7 , 3\right) \to \left(7 - 1 , 3 + 2\right) \to \left(6 , 5\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(2,7)" and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 5\right)$

$d = \sqrt{{\left(6 - 2\right)}^{2} + {\left(5 - 7\right)}^{2}} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47$

$\text{sum of radii } = 5 + 4 = 9$

$\text{since sum of radii">d" then circles overlap}$
graph{((x-2)^2+(y-7)^2-25)((x-6)^2+(y-5)^2-16)=0 [-20, 20, -10, 10]}