Circle A has a radius of 2  and a center of (7 ,6 ). Circle B has a radius of 3  and a center of (5 ,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?

May 28, 2016

circles overlap

Explanation:

What we have to do here is 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

The first step is to find the new centre of B under the given translation. Under a translation the shape of the figure does not change only it's position.

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

centre of B (5 ,3) → (5-1 ,3+2) → (4 ,5)

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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$

The 2 points here are (7 ,6) and (4 ,5)

d=sqrt((4-7)^2+(5-6)^2)=sqrt10≈3.162

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

Since sum of radii > d , then circles overlap.
graph{(y^2-12y+x^2-14x+81)(y^2-10y+x^2-8x+32)=0 [-20, 20, -10, 10]}