Question

Circle A has a center at `(2 ,3 )` and an area of `8 pi`. Circle B has a center at `(11 ,7 )` and an area of `54 pi`. Do the circles overlap?

Answer 1

`"circles overlap"`

Explanation:

`"What we have to do here is compare the distance (d)"`
`"between the centres to the sum of the radii"`

`• " if sum of radii">d" then circles overlap"`

`• " if sum of radii"< d" then no overlap"`

`"To calculate the radii use the area formula "A=pir^2`

`r_(A)=sqrt8=2sqrt2" and "r_(B)=sqrt54=3sqrt6`

`"to calculate d use the "color(blue)"distance formula"`

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(2,3)" and "(x_2,y_2)=(11,7)`

`d=sqrt((11-2)^2+(7-3)^2)=sqrt(81+16)=sqrt97~~9.85`

`"sum of radii "=2sqrt2+3sqrt6~~10.18`

`"Since sum of radii">d" then circles overlap"`
graph{((x-2)^2+(y-3)^2-(2sqrt2)^2)((x-11)^2+(y-7)^2-(3sqrt6)^2)=0 [-20, 20, -10, 10]}