Question

Circle A has a center at `(1 ,8 )` and an area of `18 pi`. Circle B has a center at `(8 ,1 )` and an area of `45 pi`. Do the circles overlap?

Answer 1

The circles overlap

Explanation:

Circle A
`area =pir_A^2=18pi`

So, `r_A=sqrt18=3sqrt2`

Circle B
`area=pir_B^2=45pi`

So, `r_B=sqrt45=3sqrt5`

The distance between the centers of the circles `A (x_A,y_A)` and `B(x_B,y_B)`

`d=sqrt((x_B-x_A)^2+(y_b-y_A)^2)`

`d=sqrt((8-1)^2+(1-8)^2)=sqrt(7^2+7^2)=7sqrt2=9.9`

The sum of the radii `=r_A+r_B=3sqrt2+3sqrt5=10.95`

Therefore, `d<=r_A+r_B`
So, the circles overlap

The equations of the circles are

`(x-1)^2+(y-8)^2=18`

`(x-8)^2+(y-1)^2 =45`
graph{((x-1)^2+(y-8)^2-18)((x-8)^2+(y-1)^2-45)=0 [-22.45, 23.15, -5.73, 17.09]}