Circle A has a center at #(5 ,4 )# and an area of #54 pi#. Circle B has a center at #(12 ,8 )# and an area of #25 pi#. Do the circles overlap?

1 Answer
Nov 8, 2016

Two circles overlap (intersect) each other.

Explanation:

Circle A has a center at #(5,4)# and its radius is #sqrt((54pi)/pi)=sqrt54=7.348#

Circle B has a center at #(12,8)# and its radius is #sqrt((25pi)/pi)=5#

Sum of the radii is #12.348# and difference is #2.348#

The distance between centers is

#sqrt((12-5)^2+(8-4)^2)=sqrt(7^2+4^2)=sqrt65=8.062#

The distance between centers at #8.062# is less than sum of radii and greater than difference in radii.

Hence, two circles intersect each other. For details see here
graph{(x^2+y^2-10x-8y-13)(x^2+y^2-24x-16y+183)=0 [-13.92, 26.08, -4.96, 15.04]}