# Circle A has a radius of 2  and a center of (2 ,5 ). Circle B has a radius of 3  and a center of (7 ,8 ). 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?

Oct 8, 2016

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

Before calculating d , we require to find the ' new' centre of B under the given translation which does not change the shape of the circle only it's position.

Under the translation $\left(\begin{matrix}- 2 \\ - 4\end{matrix}\right)$

$\left(7 , 8\right) \to \left(7 - 2 , 8 - 4\right) \to \left(5 , 4\right) \leftarrow \text{ new centre of B}$

To calculate d, use the$\textcolor{b l u e}{\text{ distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (2 ,5) and (5 ,4)

let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(5 , 4\right)$

d=sqrt((5-2)^2+(4-5)^2)=sqrt(9+1)=sqrt10≈3.162

Sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x+25)(y^2-8y+x^2-10x+32)=0 [-28.86, 28.87, -14.43, 14.43]}