Question

Circle A has a center at `(2 ,12 )` and an area of `81 pi`. Circle B has a center at `(1 ,3 )` and an area of `16 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

circles overlap

Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before doing this, we require to find the radii of both circles.

`color(orange)"Reminder"` The area (A) of a circle is

`color(red)(|bar(ul(color(white)(a/a)color(black)(A=pir^2)color(white)(a/a)|)))`

`color(blue)"Circle A " pir^2=81pirArrr^2=(81cancel(pi))/cancel(pi)rArrr=9`

`color(blue)"Circle B " pir^2=16pirArrr^2=(16cancel(pi))/cancel(pi)rArrr=4`

To calculate d, use the `color(blue)"distance formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where`(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (2 ,12) and (1 ,3) the centres of the circles.

let `(x_1,y_1)=(2,12)" and " (x_2,y_2)=(1,3)`

`d=sqrt((1-2)^2+(3-12)^2)=sqrt(1+81)=sqrt82≈9.055`

sum of radii = radius of A + radius of B = 9 + 4 = 13

Since sum of radii > d , then circles overlap
graph{(y^2-24y+x^2-4x+67)(y^2-6y+x^2-2x-6)=0 [-56.96, 56.94, -28.5, 28.46]}