Circle A has a radius of #2 # and a center of #(5 ,7 )#. Circle B has a radius of #4 # and a center of #(3 ,2 )#. 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
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 of the radii"#
#• " if sum of radii">d" then circles overlap"#
#• " if sum of radii" < d" then no overlap"# We require to 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>#
#(3,2)to(3+2,2-1)to(5,1)larrcolor(red)" new centre of B"# Note that the x-coordinate of the centres of both circles is 5 indicating that they lie on the vertical line x = 5
Hence d is the difference in the y-coordinates.
#rArrd=7-1=6#
#"sum of radii "=2+4=6#
#"since sum of radii "=d=6#
#"then the circles touch externally"#
graph{((x-5)^2+(y-7)^2-4)((x-5)^2+(y-1)^2-16)=0 [-20, 20, -10, 10]}
