Two circles have the following equations: #(x -8 )^2+(y -5 )^2= 9 # and #(x +4 )^2+(y +2 )^2= 25 #. 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?

graph{0= ( (x-8)^2 + (y-5)^2-9) (( x + 4 )^ 2 + ( y + 2 ) ^2 - 25) [-20, 20, -10,10]}

To deduce in general the maximum of the distance between any two points on two curves is a variational calculus problem, which is quite an advanced technique, and not likely to be what's wanted here.

Fortunately, there's a much simpler way to get to the answer in this case -the maximum distance is found when one draws a line through the centre of the two circles and takes its far intersection with each circle. To see that this is so, consider the triangle inequality; the logic's written out here: https://math.stackexchange.com/questions/437313/maximum-distance-between-points-on-circle

So the maximum distance is the distance between the centres of the two circles plus the radii of the two circles:

As 193 is a prime number, the root does not simplify further.

To determine if one circle contains the other, we can compare the distance between their centers and the sum of their radii.

For the first circle: Center ( (8, 5) ), radius ( r_1 = 3 ).

For the second circle: Center ( (-4, -2) ), radius ( r_2 = 5 ).

The distance between their centers can be found using the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

[ d = \sqrt{(-4 - 8)^2 + (-2 - 5)^2} = \sqrt{(-12)^2 + (-7)^2} = \sqrt{144 + 49} = \sqrt{193} ]

Now, we compare this distance with the sum of the radii:

[ r_1 + r_2 = 3 + 5 = 8 ]

Since ( \sqrt{193} > 8 ), the circles do not contain each other.

To find the greatest possible distance between a point on one circle and another point on the other circle, we need to add the radius of one circle to the distance between their centers, and vice versa. This will give us the maximum distance between any two points on the circles.

So, the greatest possible distance ( D ) is:

[ D = \sqrt{193} + 5 = 5 + \sqrt{193} ]