Question

Circle A has a radius of `2` and a center of `(6 ,3 )`. Circle B has a radius of `3` and a center of `(2 ,4 )`. If circle B is translated by `<1 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles have one point of contact.

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

#• If sum of radii = d , then one point of contact

Before calculating d, we have 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 a translation of " ((1),(3))`

`(2,4)to(2+1,4+3)to(3,7)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 (6 ,3) and (3 ,7)

let `(x_1,y_1)=(6,3)" and " (x_2,y_2)=(3,7)`

`d=sqrt((3-6)^2+(7-3)^2)=sqrt(9+16)=sqrt25=5`

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

Since sum of radii = d , then circles have one point of contact.
graph{(y^2-6y+x^2-12x+41)(y^2-14y+x^2-6x+49)=0 [-25.64, 25.68, -12.82, 12.83]}