Question

Circle A has a center at `(3 ,4 )` and an area of `16 pi`. Circle B has a center at `(8 ,1 )` and an area of `40 pi`. Do the circles overlap?

Answer 1

Yes. In fact the centre of circle A is contained in circle B.

Explanation:

The area of a circle is `pi r^2` where `r` is the radius.

So the radius of circle A is `sqrt(16) = 4` and the radius of circle B is `sqrt(40) = 2 sqrt(10) ~~ 6.325`

The distance between `(3, 4)` and `(8, 1)` is given by the distance formula as:

`sqrt((8-3)^2+(1-4)^2) = sqrt(5^2+3^2) = sqrt(25+9) = sqrt(34) ~~ 5.831`

So the point `(3, 4)` is actually contained in the circle B, being closer to `(8, 1)` than the radius of circle B.

graph{((x-3)^2+(y-4)^2-16)((x-3)^2+(y-4)^2-0.02)((x-8)^2+(y-1)^2-40)((x-8)^2+(y-1)^2-0.02) = 0 [-14.58, 25.42, -8.4, 11.6]}