Circle A has a center at #(5 ,1 )# and an area of #23 pi#. Circle B has a center at #(2 ,8 )# and an area of #15 pi#. Do the circles overlap?

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Answer 1 · 2017-02-06T21:52:12

Yes...

The area of a circle is given by the formula:

#a = pi r^2#

where #r# is the radius.

So given the area #a#, the radius is given by the formula:

#r = sqrt(a/pi)#

Hence:

The distance #d# between two points #(x_1, y_1)# and #(x_2, y_2)# is given by the formula:

#d = sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

So the distance between the centre #(x_1, y_1) = (5, 1)# of circle A and the centre #(x_2, y_2) = (2, 8)# of circle B is:

#d = sqrt((2-5)^2+(8-1)^2) = sqrt(3^2+7^2) = sqrt(9+49) = sqrt(58)#

Then we find:

#sqrt(23)+sqrt(15) ~~ 4.8+3.9 = 8.7 > 7.6 ~~ sqrt(58)#

Since the sum of the radii is greater than the distance between the centres of the circles, they do overlap...

graph{((x-5)^2+(y-1)^2-23)((x-5)^2+(y-1)^2-0.09)((x-2)^2+(y-8)^2-15)((x-2)^2+(y-8)^2-0.1) = 0 [-12, 22, -5, 12]}