Question

Circle A has a radius of `2` and a center of `(2 ,5 )`. Circle B has a radius of `3` and a center of `(7 ,8 )`. 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

circles overlap.

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 the translation `((-2),(-4))`

`(7,8)to(7-2,8-4)to(5,4)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 ,5) and (5 ,4)

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

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

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

Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x+25)(y^2-8y+x^2-10x+32)=0 [-28.86, 28.87, -14.43, 14.43]}