Question

Circle A has a radius of `3` and a center at `(1 ,3 )`. Circle B has a radius of `5` and a center at `(1 ,7 )`. If circle B is translated by `<2 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈0.246

Explanation:

What we have to do here is compare the distance (d) between the centres to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Our first step is to find the new centre of B under the given translation.The shape of a figure does not change under a translation only it's position.

Under a translation `((2),(4))`

centre of B (1 ,7) → (1+2 ,7+4) → (3 ,11)

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 points"`

Here the 2 points are (1 ,3) and (3 ,11)

`d=sqrt((3-1)^2+(11-3)^2)=sqrt68≈8.246`

radius of A + radius of B = 3 + 5 = 8

Since sum of radii < d , then no overlap.

minimum distance between 2 points = 8.246 - 8 = 0.246
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