Question

Circle A has a radius of `5` and a center of `(2 ,7 )`. Circle B has a radius of `1` and a center of `(6 ,1 )`. If circle B is translated by `<2 ,7 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , min. distance ≈ 0.082 units

Explanation:

What we have to do here is to compare the distance ( d) between the centres of the circles with 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 require to find the new coordinates of the centre of B under the translation, which does not change the shape of the circle, only it's position.

Under a translation `((2),(7))`

(6 ,1) → (6+2 ,1+7) → (8 ,8) is the new centre of circle 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 ,7) and (8 ,8) the centres of the circles.

let `(x_1,y_1)=(2,7)" and " (x_2,y_2)=(8,8)`

`d=sqrt((8-2)^2+(8-7)^2)=sqrt(3 6+1)=sqrt37≈6.082`

sum of radii = radius of A + radius of B = 5 + 1 = 6

Since sum of radii < d , then no overlap

min. distance between points = d - sum of radii

= 6.082 - 6 = 0.082 units (3 decimal paces)
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