Circle A has a center at #(2 ,12 )# and an area of #81 pi#. Circle B has a center at #(1 ,3 )# and an area of #16 pi#. Do the circles overlap? If not, what is the shortest distance between them?
1 Answer
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=81pirArrr^2=(81cancel(pi))/cancel(pi)rArrr=9#
#color(blue)"Circle B " pir^2=16pirArrr^2=(16cancel(pi))/cancel(pi)rArrr=4# 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"# The 2 points here are (2 ,12) and (1 ,3) the centres of the circles.
let
# (x_1,y_1)=(2,12)" and " (x_2,y_2)=(1,3)#
#d=sqrt((1-2)^2+(3-12)^2)=sqrt(1+81)=sqrt82≈9.055# sum of radii = radius of A + radius of B = 9 + 4 = 13
Since sum of radii > d , then circles overlap
graph{(y^2-24y+x^2-4x+67)(y^2-6y+x^2-2x-6)=0 [-56.96, 56.94, -28.5, 28.46]}