Question

Two circles have the following equations `(x +5 )^2+(y +3 )^2= 9` and `(x +4 )^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

No overlap of the circles. Greatest distance `=8.123`

Explanation:

Circle A, center (-5,-3), `r_A=3`
Circle B, center (-4,1), `r_B=1`

Now 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 the difference of the radii `>d`, then one circle inside the other.

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,-3)` and `(-4,1)` the centres of the circles

let`(x_1,y_1)=(-5,-3)` and `(x_2,y_2)=(-4,1)`

`d=sqrt((-4-(-5))^2+(1-(-3))^2)=sqrt(1^2+(4)^2)=sqrt17=4.123`

Sum of radii = radius of A + radius of B `= 3+1=4`

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

greatest distance :
`d+`sum of radii `= 4.123+3+1 =8.123`