# 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 ,1 ). If circle B is translated by <2 ,7 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jun 18, 2017

$\text{circles overlap}$

#### Explanation:

$\text{what we have to do here is " color(blue)"compare "" the}$
$\text{distance (d) between the centres to the "color(blue)"sum of the radii}$

• " if sum of radii ">d" then circles overlap"

• " if sum of radii "< d" then no overlap"

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

$\text{under a translation} \left(\begin{matrix}2 \\ 7\end{matrix}\right)$

$\left(6 , 1\right) \to \left(6 + 2 , 7 + 1\right) \to \left(8 , 8\right) \leftarrow \text{ new centre of B}$

$\text{to calculate d note the centres are " (8,5)" and } \left(8 , 8\right)$

$\text{the x-coordinates are equal so centres lie on a }$
$\text{vertical line and }$

$d = 8 - 5 = 3$

$\text{sum of radii } = 4 + 2 = 6$

$\text{since sum of radii ">d" then circles overlap}$
graph{(y^2-16y+x^2-16x+124)(y^2-10y+x^2-16x+73)=0 [-20, 20, -10, 10]}