Question

Two circles have the following equations `(x +5 )^2+(y +6 )^2= 9` and `(x +2 )^2+(y -1 )^2= 81`. 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

The circles intersect but neither one of them contains the other.
Greatest possible distance `color(blue)(d_f=19.615773105864" "`units

Explanation:

The given equations of the circle are
`(x+5)^2+(y+6)^2=9" "`first circle
`(x+2)^2+(y-1)^2=81" "`second circle

We start with the equation passing thru the centers of the circle

`C_1(x_1, y_1)=(-5, -6)` and `C_2(x_2, y_2)=(-2, 1)` are the centers.

Using two-point form

`y-y_1=((y_2-y_1)/(x_2-x_1))*(x-x_1)`

`y--6=((1--6)/(-2--5))*(x--5)`

`y+6=((1+6)/(-2+5))*(x+5)`

`y+6=((7)/(3))*(x+5)`

After simplification

`3y+18=7x+35`

`7x-3y=-17" "`equation of the line passing thru the centers and at the two points farthest to each other.

Solve for the points using first circle and the line
`(x+5)^2+(y+6)^2=9" "`first circle
`7x-3y=-17" "`the line

One point at `A(x_a, y_a)=(-6.1817578957376, -8.7574350900543)`
Another at `B(x_b, y_b)=(-3.8182421042626, -3.2425649099459)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Solve for the points using second circle and the line
`(x+2)^2+(y-1)^2=81" "`second circle
`7x-3y=-17" "`the line

One point at `C(x_c, y_c)=(1.5452736872127, 9.2723052701629)`
Another at `D(x_d, y_d)=(-5.5452736872127, -7.2723052701625)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To compute for the farthest distance `d_f` we will use point `A` and `C`

`d_f=sqrt((x_a-x_c)^2+(y_a-y_c)^2)`

`d_f=sqrt((-6.1817578957376-1.5452736872127)^2+(-8.7574350900543-9.2723052701629)^2)`

`color(blue)(d_f=19.615773105864" "`unit)s

Kindly see the graph

God bless .... I hope the explanation is useful.