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?

Question

1570 views around the world

You can reuse this answer
Creative Commons License

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.

enter image source here

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#