Question

Circle A has a radius of `4` and a center of `(6 ,1 )`. Circle B has a radius of `1` and a center of `(5 ,3 )`. If circle B is translated by `<-3 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap, ≈ 0.657

Explanation:

What we have 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

The first step, however , is 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 `((-3),(2))`

`(5,3)to(5-3,3+2)to(2,5)" 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"`

The 2 points here are (6 ,1) and (2 ,5) the centres of the circles.

let `(x_1,y_1)=(6,1)" and " (x_2,y_2)=(2,5)`

`d=sqrt((2-6)^2+(5-1)^2)=sqrt(16+16)=sqrt32≈5.657`

sum of radii = radius of A + radius of B = 4 + 1 = 5

Since sum of radii < d , then no overlap and min. distance between circles is.

min. distance = d - sum of radii = 5.657 - 5 = 0.657 units.
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