Question

Circle A has a center at `(5 ,2 )` and an area of `15 pi`. Circle B has a center at `(2 ,3 )` and an area of `75 pi`. Do the circles overlap?

Answer 1

Smaller circle lies inside the larger circle and the two overlap.

Explanation:

Whether the two circles with given centers and radii intersect each other, touch each other (internally or externally) are separated and do not overlap or bigger circle engulfs smaller circle, critically depends on the distance between the centers and sum / difference of their radii . One can find the details here.

Here the distance between centers `(5,2)` and `(2,3)` is `sqrt((2-5)^2+(3-2)^2)=sqrt(9+1)=sqrt10=3.1623`.

As areas of circle are given by `pir^2`, radius is `sqrt((Area)/pi)` and while radius of first circle with center `(5,2)` is `sqrt((15pi)/pi)=sqrt15=3.873` and radius of second circle with center `(2,3)` is `sqrt((75pi)/pi)=sqrt75=8.6603`.

Sum of radii is `8.6603+3.873=12.5333` and difference of radii is `8.6603-3.873=4.7873`.

As sum of radii is greater than distance between centers and difference between radii is also greater than distance between centers, smaller circle lies inside the larger circle and the two overlap.
graph{ (x^2+y^2-10x-4y+14)(x^2+y^2-4x-6y-62)=0 [-19.33, 20.67, -7.36, 12.64]}