Circle A has a radius of 2  and a center of (2 ,6 ). Circle B has a radius of 3  and a center of (7 ,8 ). If circle B is translated by <-2 ,-3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Mar 25, 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 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"

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

$\left(7 , 8\right) \to \left(7 - 2 , 8 - 3\right) \to \left(5 , 5\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

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

$d = \sqrt{{\left(2 - 5\right)}^{2} + {\left(6 - 5\right)}^{2}} = \sqrt{9 + 1} = \sqrt{10} \approx 3.16$

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

$\text{Since sum of radii ">d" then circles overlap}$
graph{((x-2)^2+(y-6)^2-4)((x-5)^2+(y-5)^2-9)=0 [-20, 20, -10, 10]}