Circle A has a radius of #5 # and a center of #(3 ,2 )#. Circle B has a radius of #3 # and a center of #(1 ,4 )#. If circle B is translated by #<2 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Nov 20, 2016

circle B inside circle A

Explanation:

What we have to do here is #color(blue)"compare"# the distance ( d) between the centres of the circles to the #color(blue)"sum/difference of the radii"#

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

• If difference of radii > d , then one circle inside the other

Before calculating d we must find the ' new' centre of B under the given translation which does not change the shape of the circle only it's position.

Under a translation #((2),(-1))#

#(1,4)to(1+2,4-1)to(3,3)larr" new centre of B"#

Since the centres (3 ,2) and (3 ,3) have the same x-coordinate they lie on the vertical line with equation x = 3 and d is the difference of the y-coordinates.

#rArrd=3-2=1#

difference of radii= radius of A - radius of B = 5 - 3 = 2

Since difference of radii > d , then circle B is inside circle A.
graph{(y^2-4y+x^2-6x-12)(y^2-6y+x^2-6x+9)=0 [-20, 20, -10, 10]}