Circle A has a center at #(1 ,4 )# and a radius of #5 #. Circle B has a center at #(9 ,3 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

2 Answers
May 3, 2016

There is a minimum distance of #(sqrt(65)-6)~~2.06# units between the two (non-overlapping) circles.

Explanation:

The distance between the two centers is
#color(white)("XXX")d=sqrt((9-1)^2+(4-3)^2)=sqrt(65)~~8.06#

Along the line segment connecting the two centers
#5# units are covered by circle A, and
#1# unit is covered by circle B.

So only #5+1=6# units are covered by the circles.

Leaving #sqrt(65)-6~~2.06# units uncovered.

May 3, 2016

no overlap , ≈ 2.06

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

To calculate the distance between the centres 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 points"#

let # (x_1,y_1)=(1,4)" and " (x_2,y_2)=(9,3)#

#d=sqrt((9-1)^2+(3-4)^2)=sqrt65 ≈ 8.06#

radius of A + radius of B = 5 + 1 = 6

Since sum of radii < d , then no overlap

smallest distance = 8.06 - 6 = 2.06
graph{(y^2-8y+x^2-2x-8)(y^2-6y+x^2-18x+89)=0 [-35.56, 35.56, -17.78, 17.78]}