Question

Circle A has a radius of `4` and a center of `(8 ,5 )`. Circle B has a radius of `2` and a center of `(6 ,7 )`. If circle B is translated by `<3 ,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 with 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 `((3),(1))`

`(6,7)to(6+3,7+1)to(9,8)larrcolor(blue)" 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 (8 ,5) and (9 ,8)

let `(x_1,y_1)=(8,5)" and " (x_2,y_2)=(9,8)`

`d=sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10≈3.162`

sum of radii = radius of A + radius of B = 4 + 2 = 6

Since sum of radii > d , then circles overlap.
graph{(y^2-10y+x^2-16x+73)(y^2-16y+x^2-18x+141)=0 [-25.31, 25.32, -12.66, 12.65]}