Circle A has a radius of #3 # and a center of #(3 ,3 )#. Circle B has a radius of #5 # and a center of #(1 ,7 )#. 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
Jul 23, 2016
circles overlap.
Explanation:
What we have to do here is compare the distance (d ) between the centres with the sum of the radii.
• If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
The first step is to calculate the 'new' coordinates of the centre of circle B under the given translation. Note the circle remains a circle but it's position changes.
Under a translation
#((2),(-1))# (1 ,7) → (1+2 ,7-1) → B(3 ,6)
To calculate d , note that the centres A(3 ,3) and B(3 ,6) have the same x-coordinate and so d is just the difference in the y-coordinates.
Hence d = 6 - 3 = 3
Sum of radii = radius of A + radius of B = 3 + 5 = 8
Since sum of radii > d , then circles overlap.
graph{(y^2-6y+x^2-6x+9)(y^2-12y+x^2-6x+20)=0 [-40, 40, -20, 20]}
