Circle A has a radius of 4  and a center of (8 ,5 ). Circle B has a radius of 2  and a center of (6 ,7 ). If circle B is translated by <3 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jan 6, 2017

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 with 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 $\left(\begin{matrix}3 \\ 1\end{matrix}\right)$

$\left(6 , 7\right) \to \left(6 + 3 , 7 + 1\right) \to \left(9 , 8\right) \leftarrow \textcolor{b l u e}{\text{ new centre of B}}$

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

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

The 2 points here are (8 ,5) and (9 ,8)

let $\left({x}_{1} , {y}_{1}\right) = \left(8 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(9 , 8\right)$

d=sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10≈3.162#

sum of radii = radius of A + radius of B = 4 + 2 = 6

Since sum of radii > d , then circles overlap.
graph{(y^2-10y+x^2-16x+73)(y^2-16y+x^2-18x+141)=0 [-25.31, 25.32, -12.66, 12.65]}