Question

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

Answer 1

no overlap , ≈ 0.162

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 `((1),(-4))`

`(4,7)to(4+1,7-4)to(5,3)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)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (2 ,4) and (5 ,3)

let `(x_1,y_1)=(2,4)" and " (x_2,y_2)=(5,3)`

`d=sqrt((5-2)^2+(3-4)^2)=sqrt(9+1)=sqrt10≈3.162`

Sum of radii = 1 + 2 = 3

Since sum of radii < d, then there is no overlap

min. distance between points = d - sum of radii

`=3.162-3=0.162" to 3 decimal places"`
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