Question

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

Answer 1

no overlap , min. distance ≈1.083 units.

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 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 the translation `((-1),(2))`

`(2,3)to(2-1,3+2)to(1,5)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 ,6) and (1 ,5)

let `(x_1,y_1)=(7,6)" and " (x_2,y_2)=(1,5)`

`d=sqrt((1-7)^2+(5-6)^2)=sqrt(36+1)=sqrt37≈6.083`

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

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

min. distance between points = d - sum of radii

`=6.083-5=1.083`
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