Question

Circle A has a center at `(4 ,-8 )` and a radius of `3`. Circle B has a center at `(-2 ,-2 )` and a radius of `2`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

The circles do not overlap and the shortest distance is `=3.5`

Explanation:

The distance between the centers is

`O_AO_B=sqrt((-2-(4))^2+(-2-(-8))^2)`

`=sqrt(36+36)`

`=sqrt72=8.5`

The sum of the radii is

`r_A+r_B=3+2=5`

As,

`O_AO_B> (r_A+r_B)`

The circles do not overlap.

The smallest distance is

`d=8.5-5=3.5`

graph{((x-4)^2+(y+8)^2-9)((x+2)^2+(y+2)^2-4)(y+x+4)=0 [-25.84, 25.46, -16.57, 9.1]}

Answer 2

`"no overlap " d~~3.485`

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"`

`"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)|)))`
`(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

`"the points are " (x_1,y_1)=(4,-8),(x_2,y_2)=(-2,-2)`

`d=sqrt((-2-4)^2+(-2+8)^2)=sqrt72~~8.485`

`"sum of radii "=3+2=5`

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

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

`=8.485-5=3.485`
graph{(y^2+16y+x^2-8x+71)(y^2+4y+x^2+4x+4)=0 [-25.31, 25.32, -12.66, 12.65]}