Question

Circle A has a radius of `2` and a center at `(7 ,1 )`. Circle B has a radius of `1` and a center at `(3 ,2 )`. If circle B is translated by `<-2 ,6 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap, min. distance ≈ 6.22

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 B under the given translation which does not change the shape of the circle only it's position.

Under a translation `((-2),(6))`

`(3,2)to(3-2,2+6)to(1,8)larr" new centre of B"`

To calculate d, use the `color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 points here are (7 ,1) and (1 ,8)

let `(x_1,y_1)=(7,1)" and " (x_2,y_2)=(1,8)`

`d=sqrt((1-7)^2+(8-1)^2)=sqrt(36+49)=sqrt85≈9.22`

sum of radii = radius of A + radius of B = 2+1 = 3

Since sum of radii < d , then circles do not overlap.

min. distance between points = d - sum of radii

`rArr"min. distance " =9.22-3=6.22`
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