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

May 12, 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 the sum of radii > d , then circles overlap

• If the sum of radii < d , then no overlap

Firstly we require to find the coordinates of the centre of B under the given translation.

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

centre B (7 ,3) → (7-3 ,3+2) → (4 ,5)

To calculate the distance (d) between the centres 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 $\left({x}_{1}\right) \text{ and " (x_2,y_2)" are 2 coord points}$

let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 6\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 5\right)$

d=sqrt((4-2)^2+(5-6)^2)=sqrt(4+1)=sqrt5 ≈ 2.236

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

Since sum of radii > d , then circles overlap
graph{(y^2-12y+x^2-4x+31)(y^2-10y+x^2-8x+25)=0 [-15.59, 15.59, -7.8, 7.79]}