Question

Circle A has a center at `(4 ,-1 )` and a radius of `5`. Circle B has a center at `(-3 ,6 )` and a radius of `2`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

They don't overlap. The closest they get is `7sqrt2 - 7`.

Explanation:

The radii of the two circles are `2` and `5`, so the circles will touch or overlap if the distance between the centres is `7` or less.
We can work out the distance between the centres with Pythagoras. The distance in x is `4--3=7`, and the distance in y is `6--1=7`, so the distance between the two centres is `sqrt{7^2 + 7^2} = 7sqrt2`. As the distance between the two centres is more than the sum of the two radii, they cannot touch.
The closest distance, therefore, will be `7sqrt2 - 7`, the distance between the centres minus the radii.

Answer 2

`"no overlap",~~2.899`

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 radii"`

`• " if sum of radii" > d" then circles overlap"`

`• " if sum of radii"< d" then no overlap"`

`"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 are " (x_1,y_1)=(4,-1),(x_2,y_2)=(-3,6)`

`d=sqrt((-3-4)^2+(6+1)^2)=sqrt(49+49)=sqrt98~~9.899`

`"sum of radii "=5+2=7`

`"Since sum of radii"< d" then no overlap"`

`"smallest distance "=d-" sum of radii"`

`color(white)(smallest distance)=9.899-7`

`color(white)(smallest distance)=2.899`
graph{(y^2+2y+x^2-8x-8)(y^2-12y+x^2+6x+41)=0 [-20, 20, -10, 10]}