# Circle A has a radius of 2  and a center at (5 ,6 ). Circle B has a radius of 5  and a center at (2 ,4 ). If circle B is translated by <-2 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

May 3, 2016

circles overlap

#### Explanation:

What we have to do here is to compare the distance (d) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

However we require to find the new centre of B under the translation.
A translation does not change the shape of a figure only it's position.

Under a translation $\left(\begin{matrix}- 2 \\ 1\end{matrix}\right)$

centre B(2 ,4) → (-2+2 ,4+1) → (0 ,5)

To calculate the distance (d) between centres use the $\textcolor{b l u e}{\text{ distance formula }}$

color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|))
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$

let $\left({x}_{1} , {y}_{1}\right) = \left(5 , 6\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(0 , 5\right)$

rArr d=sqrt((0-5)^2+(5-6)^2)=sqrt26 ≈ 5.099

radius of A + radius of B = 2 + 5 = 7

Since sum of radii > d , then circles overlap
graph{(y^2-12y+x^2-10x+57)(y^2-10y+x^2)=0 [-20, 20, -10, 10]}