Question

Circle A has a center at `(3 ,1 )` and an area of `15 pi`. Circle B has a center at `(5 ,2 )` and an area of `24 pi`. Do the circles overlap?

Answer 1

Circles intersect each other and greatest possible distance between a point on one circle and another point on the other is `10.908`.

Explanation:

Please refer to details here.

The center of first circle is `(3,1)` and as area is `15pi`, the radius is `sqrt15=3.873` (as `pir^2=15pi`, `r=sqrt15`) and center of second circle is `(5,2)` and radius is `sqrt24=4.899`.

The distance between centers is `sqrt((5-3)^2+(2-1)^2)`

= `sqrt(4+1)=sqrt5=2.236`

Let the radii of two circles is `r_1` and `r_2` and we also assume that `r_1>r_2` and the distance between centers is `d`.

So here `r_1+r_2=8.772 > d=2.236` and `r_1-r_2=1.026 < d=2.236`

Hence, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is `3.873+4.899+2.236=10.908`

graph{(x^2+y^2-6x-2y-5)(x^2+y^2-10x-4y+5)=0 [-6.25, 13.75, -2.92, 7.08]}