Question

Circle A has a center at `(3 ,4 )` and an area of `64 pi`. Circle B has a center at `(1 ,12 )` and an area of `54 pi`. Do the circles overlap?

Answer 1

The circles will intersect at two points.

Explanation:

Area of circle `A` is `A_A=pi *r_A^2= 64 pi :. r_A=8`

Area of circle `B` is `A_B=pi *r_B^2= 54 pi`

`:. r_B=sqrt 54~~7.35 (2dp) :. r_A+r_B= 15.35`

and `|r_A-r_B| = 0.65`

Center of first circle `A` is at `(3,4)` and radius is `8` unit .

Center of second circle `B` is at `(1,12)` and radius is

`sqrt 54` unit . Distance between their centres is

`d=sqrt((x_1-x_2)^2+(y_1-y_2)^2)=sqrt((3-1)^2+(4-12)^2)` or

`d=sqrt(4+64)=sqrt 68 ~~ 8.25` unit.

Two circles intersect if, and only if, the distance between their

centers is between the sum and the difference of their radii.

Here `0.65 < 8.25 <15.35`, so they intersect at two points. [Ans]