Question

Circle A has a radius of `3` and a center of `(2 ,1 )`. Circle B has a radius of `2` and a center of `(7 ,3 )`. If circle B is translated by `<4 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 4.849

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 of circles

Before calculating d, we require 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 of `((4),(2))`

`7,3)to(7+4,3+2)to(11,5)larr" 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 (2 ,1) and (11 ,5)

let `(x_1,y_1)=(2,1)" and " (x_2,y_2)=(11,5)`

`d=sqrt((11-2)^2+(5-1)^2)=sqrt(81+16)=sqrt97≈9.849`

Sum of radii = radius of A + radius of B = 3 + 2 = 5

Since sum of radii < d , then no overlap

min. distance between circles = d - sum of radii

`=9.849-5=4.849`
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