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

Jul 11, 2016

circles overlap

Explanation:

What we require here is to compare the distance (d) between the centres to 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 coordinates of B under the translation. A translation does not change the shape of the circle only it's position.

Under a translation $\left(\begin{matrix}- 3 \\ 2\end{matrix}\right)$

B(7 ,3) → (7-3 ,3+2) → B(4 ,5) is new position of centre 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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

Here the 2 points are A(2 ,1) and B(4 ,5)

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

d=sqrt((4-2)^2+(5-1)^2)=sqrt(4+16)=sqrt20≈4.472

Sum of radii = radius of A + radius of B = 3 + 4 = 7

Since sum of radii > d , then circles overlap
graph{(y^2-2y+x^2-4x-5)(y^2-10y+x^2-8x+25)=0 [-20, 20, -10, 10]}