Circle A has a radius of #6 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(6 ,7 )#. 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?
1 Answer
circles overlap.
Explanation:
What we require to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.
• If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
Before doing this we need 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 the translation
#((3),(1))# (6 ,7) → (6+3 ,7+1) → (9 ,8) is the new centre of B.
To calculate d, use the
#color(blue)"distance formula"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where# (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# Here the 2 points are (2 ,5) and (9 ,8) the centres of the circles.
let
# (x_1,y_1)=(2,5)" and " (x_2,y_2)=(9,8)#
#d=sqrt((9-2)^2+(8-5)^2)=sqrt(49+9)=sqrt58≈7.616# sum of radii = radius of A + radius of B = 6 + 3 = 9
Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x-7)(y^2-16y+x^2-18x+136)=0 [-40, 40, -20, 20]}
