Question

Circle A has a center at `(5 ,4 )` and an area of `15 pi`. Circle B has a center at `(2 ,1 )` and an area of `90 pi`. Do the circles overlap?

Answer 1

`"one circle inside other"`

Explanation:

What we have to do here is `color(blue)"compare "` the distance (d) between the centres of the circles to the `color(blue)"sum/difference"` of the radii.

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

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

`• " if difference of radii">d" then circle inside other"`

`• " area of circle "=pir^2larr" r is the radius"`

`color(blue)"Circle A"`

`rArrpir^2=15pirArrr=sqrt15~~3.873`

`color(blue)"Circle B"`

`rArrpir^2=90pirArrr=sqrt90~~ 9.486`

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`

`"let "(x_1,y_1)=(2,1)" and "(x_2,y_2)=(5,4)`

`d=sqrt((5-2)^2+(4-1)^2)=sqrt18~~ 4.243`

`"sum of radii "=3.873+9.486=13.359`

`"difference of radii "=9.486-3.873=5.613`

`"since difference of radii">d" then circle inside other"`
graph{((x-5)^2+(y-4)^2-15)((x-2)^2+(y-1)^2-90)=0 [-40, 40, -20, 20]}