Question

Circle A has a center at `(8 ,5 )` and an area of `100 pi`. Circle B has a center at `(4 ,2 )` and an area of `36 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 the 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=100pirArrr^2=(100cancel(pi))/cancel(pi)rArrr=10`

`color(blue)"Circle B "pir^2=36pirArrr^2=(36cancel(pi))/cancel(pi)rArrr=6`

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 (8 ,5) and (4 ,2) the centres of the circles.

`d=sqrt((8-4)^2+(5-2)^2)=sqrt(16+9)=sqrt25=5`

sum of radii = radius of A + radius of B = 10 + 6 = 16

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-8x-16)(y^2-10y+x^2-16x-11)=0 [-9.86, 9.88, -4.93, 4.935]}