Circle A has a radius of 3 3 and a center of (2 ,6 )(2,6). Circle B has a radius of 4 4 and a center of (7 ,3 )(7,3). If circle B is translated by <-3 ,2 ><−3,2>, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
circles overlap
Explanation:
What we have to do here is to compare the distance (d) between the centres with the sum of the radii.
• If the sum of radii > d , then circles overlap
• If the sum of radii < d , then no overlap
Firstly we require to find the coordinates of the centre of B under the given translation.
Under a translation of
((-3),(2)) centre B (7 ,3) → (7-3 ,3+2) → (4 ,5)
To calculate the distance (d) between the centres 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)" and " (x_2,y_2)" are 2 coord points" let
(x_1,y_1)=(2,6)" and " (x_2,y_2)=(4,5)
d=sqrt((4-2)^2+(5-6)^2)=sqrt(4+1)=sqrt5 ≈ 2.236 radius of A + radius of B = 3 + 4 = 7
Since sum of radii > d , then circles overlap
graph{(y^2-12y+x^2-4x+31)(y^2-10y+x^2-8x+25)=0 [-15.59, 15.59, -7.8, 7.79]}
