Question

Circle A has a center at `(11 ,5 )` and an area of `100 pi`. Circle B has a center at `(4 ,9 )` and an area of `36 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

circles overlap

Explanation:

First step is to find the distance between the centres using the `color(blue)" distance formula "`

`d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

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

let `(x_1,y_1)=(11,5)" and "(x_2,y_2)=(4,9)`

substitute these values into the formula to find d

`d = sqrt((4-11)^2 + (9-5)^2) = sqrt(49+16) = sqrt65 ≈ 8.06`

Now, require to find the radii of circles A and B, given that the areas are known.

Circle A : `pir^2 = 100pi rArr r^2 = (100pi)/pi = 100 rArr r = 10`

Circle B: `pir^2 = 36pi rArr r^2 = 36 rArr r = 6`

radius of circle A + radius of circle B = 10 + 6 = 16

since sum of radii > distance between centres → overlap