# Circle A has a radius of 2  and a center of (4 ,5 ). Circle B has a radius of 6  and a center of (7 ,9 ). 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?

Jun 1, 2016

1 circle enclosed inside the other.

#### Explanation:

What we have to do here is to compare the distance (d ) between the centres with the sum and difference of the radii

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

• If d < difference of radii , then circle inside other

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

Under a translation $\left(\begin{matrix}- 2 \\ - 4\end{matrix}\right)$

B (7 ,9) → (7-2 ,9-4) → (5 ,5) new centre of B

Since the centres lie on the same horizontal line (4 ,5) and (5 ,5) have the same y-coordinates then d = difference in their x-coordinates.

hence d = 5 - 4 = 1

now, radius of A + radius of B = 2 +6 = 8

and larger radius - smaller radius = 6 - 4 = 2

Since d < difference of radii , then circle inside other

graph{(y^2-10y+x^2-8x+37)(y^2-10y+x^2-10x+14)=0 [-40, 40, -20, 20]}