Circle A has a radius of #3 # and a center of #(3 ,9 )#. Circle B has a radius of #2 # and a center of #(1 ,4 )#. If circle B is translated by #<3 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2016-09-17T03:23:44

No overlap between Circle A and Circle B. Minimum distance = 1.083. See explanation and the diagram below.

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Compare the distance (d) between the centres of the circles to the sum of the radii.

1) If the sum of the radii #>#d, the circles overlap.
2) If the sum of the radii #<#d, the no overlap.

The first step here is to calculate the new centre of B under the traslation, which does not change the shape of the circle only its' position.

Under a translation #<3, -1>#
#B(1,4) => (1+3,4-1) => (4,3)# new centre of B

To calculate d, use the distance formula :
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

where #(x_1,y_1) and (x_2,y_2)# are 2 coordinate points

here the two points are #(3,9)# and #(4,3)# the centres of the circles

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

#d=sqrt((4-3)^2+(3-9)^2)=sqrt(1^2+(-6)^2)=sqrt37=6.083#

Sum of radii = radius of A + radius of B #= 3+2=5#

Since sum of radius #<#d, then no overlap of the circles

Min.D minimum distance (the red line in the daigram) :

#d-#sum of radii #= 6.083-5=1.083#