Circle A has a center at #(12 ,9 )# and an area of #25 pi#. Circle B has a center at #(3 ,1 )# and an area of #67 pi#. Do the circles overlap?

1 Answer
Aug 7, 2016

circles overlap.

Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before doing this, we require to find the radii of both circles.

#color(orange)" Reminder" # the area(A) of a circle is

#color(red)(|bar(ul(color(white)(a/a)color(black)(A=pir^2)color(white)(a/a)|)))#

#color(blue)"Circle A " pir^2=25pirArrr^2=(25cancel(pi))/cancel(pi)rArrr=5#

#color(blue)"Circle B " pir^2=67pirArrr^2=67rArrr=sqrt67≈8.185#

To calculate d, use the #color(blue)"distance formula"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"#

Here the 2 points are (12 ,9) and (3 ,1) the centres of the circles.

let # (x_1,y_1)=(12,9)" and " (x_2,y_2)=(3,1)#

#d=sqrt((3-12)^2+(1-9)^2)=sqrt(81+64)=sqrt145≈12.042#

sum of radii = radius of A + radius of B = 5 + 8.185 = 13.185

Since sum of radii > d , then circles overlap
graph{(y^2-18y+x^2-24x+200)(y^2-2y+x^2-6x-57)=0 [-41.1, 41.1, -20.55, 20.55]}