Question

Circle A has a radius of `2` and a center at `(1 ,2 )`. Circle B has a radius of `5` and a center at `(3 ,4 )`. If circle B is translated by `<2 ,1 >`, 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 doing this we require to find the new centre of circle B under the given translation, which does not change the shape of the circle, only it's position.

Under translation `((2),(1))`

`(3,4)to(3+2,4+1)to(5,5)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))|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

here the 2 points are (1 ,2) and (5 ,5) the centres of the circles.

let `(x_1,y_1)=(1,2)" and " (x_2,y_2)=(5,5)`

`d=sqrt((5-1)^2+(5-2)^2)=sqrt(16+9)=sqrt25=5`

Sum of radii = radius of A + radius of B = 2 + 5 = 7

Since sum of radii > d , then circles overlap.
graph{(y^2-4y+x^2-2x+1)(y^2-10y+x^2-10x+25)=0 [-22.5, 22.5, -11.25, 11.25]}