Circle A has a radius of 1  and a center at (3 ,3 ). Circle B has a radius of 3  and a center at (1 ,7 ). 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?

Nov 5, 2017

$\text{no overlap}$

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"

$\text{before calculating d we require to find the coordinates}$
$\text{of the centre of circle B under the given translation}$
$\text{which does not change the shape of the circle only}$
$\text{it's position}$

$\text{under the translation } < 2 , 4 >$

$\left(1 , 7\right) \to \left(1 + 2 , 7 + 4\right) \to \left(3 , 11\right) \leftarrow \textcolor{b l u e}{\text{new centre of B}}$

$\text{note that the coordinates of the 2 centres have the same}$
$\text{x-coordinate and so lie on the vertical line } x = 3$

$\text{Hence d is the difference in the y-coordinates}$

$\Rightarrow d = 11 - 3 = 8$

$\text{sum of radii } = 1 + 3 = 4$

$\text{since sum of radii"< d" then no overlap of circles}$
graph{((x-3)^2+(y-3)^2-1)((x-3)^2+(y-11)^2-9)=0 [-40, 40, -20, 20]}