Question

Circle A has a radius of `6` and a center of `(8 ,5 )`. Circle B has a radius of `3` 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 to the `color(blue)"sum of their 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 give 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(red)" 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"`

`"2 points here are " (x_1,y_1)=(8,5),(x_2,y_2)=(9,8)`

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

`"sum of radii "=6+3=9`

`"Since sum of radii" > d" then circles overlap"`
graph{(y^2-10y+x^2-16x+53)(y^2-16y+x^2-18x+136)=0 [-22.8, 22.81, -11.4, 11.4]}