Question

Two circles have the following equations `(x +5 )^2+(y +6 )^2= 9` and `(x +2 )^2+(y -1 )^2= 1`. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Answer 1

One circle does not contain the other. Greatest distance `= 11.6158.`

Explanation:

Compare the distance (d) between the centres of the circles to the sum of the radii.

1) If the sum of the radii `>`d, the circles overlap.
2) If the sum of the radii `<`d, then no overlap.
3) If `d+r_B<= r_A`, then Circle A contains Circle B

Given Circle A, centre `(-5,-6)` and radius `r_A=3`
Circle B, centre `(-2,1),` and radius `r_B=1`

The first step here is to calculate d, use the 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

here the two points are `(-5,-6)` and `(-2,1)` the centres of the circles

let`(x_1,y_1)=(-5,-6)` and `(x_2,y_2)=(-2,1)`

`d=sqrt(-2-(-5)^2+(1-(-6)^2)`
`=sqrt(3^2+7^2)=sqrt58=7.6158`

Sum of radii = radius of A `(r_A)`+ radius of B `(r_B)` `= 3+1=4`

Since sum of radius `<`d, then no overlap of the circles
no overlap => no containment

Greatest distance = `d`(the yellow segment) `+r_A+r_B`

`= 7.6158+3+1=11.6158`