Question

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

Answer 1

The distance between the centres is given by:

`r = sqrt((y_2-y_1)^2+(x_2-x_1)^2) = 8.54` units

Since this is larger than 3 units, the sum of the radii, the circles do not overlap.

Explanation:

The sum of the two radii is `2+1=3` units. If the centres are less than `3` units apart, the circles overlap, if they are more than `3` units apart then they do not.

The distance between the centres is given by:

`r = sqrt((y_2-y_1)^2+(x_2-x_1)^2)`

`r = sqrt((-5-3)^2+(-2-1)^2) = sqrt((-8)^2+(-3)^2)`

`:. r = sqrt(64+9) = sqrt(73) = 8.54` units

Since this is larger than 3 units, the circles do not overlap, and this is the distance between their centres.

Answer 2

`"no overlap, min. distance" "~~5.544`

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"`

`"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 points are " (x_1,y_1)=(1,3),(x_2,y_2)=(-2,-5)`

`d=sqrt((-2-1)^2+(5-3)^2)=sqrt(9+64)=sqrt73~~8.544`

`color(blue)"sum of radii "=1+2=3`

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

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

`rArr"smallest distance "=8.544-3=5.544`
graph{(y^2-6y+x^2-2x+9)(y^2+10y+x^2+4x+25)=0 [-12.49, 12.48, -6.24, 6.25]}