Question

Circle A has a center at `(1 ,4 )` and an area of `28 pi`. Circle B has a center at `(7 ,2 )` and an area of `8 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

circles overlap

Explanation:

Require to find the radii of circles and the distance between their centres.

Calculate distance between centres using `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)=(1,4)" and " (x_2,y_2)=(7,2)`

`rArr d=sqrt((7-1)^2+(2-4)^2 )= sqrt(36+4)=sqrt40 ≈ 6.325`
`color(red)"--------------------------------------------------------------"`
To calculate radii , use area of circle `= pir^2`

circle A : `pir^2 = 28pi rArr r^2 = 28 " and " r = sqrt28 ≈ 5.292`

circle B : `pir^2 = 8pi rArr r^2= 8" and " r=sqrt8 ≈ 2.828`

now: radius A + radius B > distance between centres

hence circles overlap