Question

Circle A has a center at `(2 ,5 )` and an area of `16 pi`. Circle B has a center at `(8 ,1 )` and an area of `3 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

`"no overlap "~~1.479`

Explanation:

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

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

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

`"to find the radii use area (A) of circle formula"`

`•color(white)(x)A=pir^2`

`"circle A "to pir^2=16pirArrr=4`

`"circle B "to pir^2=3pirArrr=sqrt3`

`"calculate d using 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)=(2,5)" and "(x_2,y_2)=(8,1)`

`d=sqrt((8-2)^2+(1-5)^2)=sqrt(36+16)=sqrt52~~7.211`

`"sum of radii "=4+sqrt3~~5.732`

`"since sum of radii"< d" then no overlap"`

`"minimum distance "=d-" sum of radii"`

`color(white)(xxxxxxxxxxxxxx)=7.211-5.732=1.479`
graph{((x-2)^2+(y-5)^2-16)((x-8)^2+(y-1)^2-3)=0 [-10, 10, -5, 5]}