Question

Circle A has a center at `(2 ,12 )` and an area of `81 pi`. Circle B has a center at `(1 ,3 )` and an area of `36 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

`"circles overlap"`

Explanation:

`"what we have to do here is to "color(blue)"compare"`
`"the distance (d) between the centres of the circles"`
`"with the "color(blue)"sum of the radii"`

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

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

`"to calculate the radii of the circles use"`

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

`color(blue)"circle A"color(white)(x)pir^2=81pirArrr=9`

`color(blue)"circle B "pir^2=36pirArrr=6`

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

`"here "(x_1,y_1)=(1,3)" and "(x_2,y_2)=(2,12)`

`d=sqrt((2-1)^2+(12-3)^2)=sqrt(1+81)=sqrt82~~ 9.055`

`"sum of radii "=9+6=15`

`"since sum of radii ">d" then circles overlap"`
graph{((x-2)^2+(y-12)^2-81)((x-1)^2+(y-3)^2-36)=0 [-40, 40, -20, 20]}