Question

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

Answer 1

no overlap, ≈ 2.602

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

Before calculating d, we require to find the ' new ' centre of circle B under the given translation which does not change the shape of the circle, only it's position.

Under a translation `((-3),(4))`

`(4,5)to(4-3,5+4)to(1,9)larr" 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"`

The 2 points here are (6 ,2) and (1 ,9)

let `(x_1,y_1)=(6,2)" and " (x_2,y_2)=(1,9)`

`d=sqrt((1-6)^2+(9-2)^2)=sqrt(25+49)=sqrt74≈8.602`

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

Since sum of radii < d , then no overlap of circles

min. distance between them = d - sum of radii

`=8.602-6=2.602`
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