Question

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?

Answer 1

Consider the line passing through the middle of the two circles. Explicit answer below.

Explanation:

We can immediately deduce that the two circles do not even overlap, let alone have one contain the other - the first has radius 3 and is centred at `(8,5)`; the second has radius 5 and is centred at `(-4,-2)`. So the minimum `x` of circle 1 is at `x=5` and the maximum `x` of circle 2 is at `x=1`.

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:

`r_1+d+r_2=`
`3+sqrt((8--4)^2+(5--2)^2)+5=`
`3+sqrt(12^2+7^2)+5=`
`3+sqrt(144+49)+5=`
`8+sqrt(193)`

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

Answer 2

The distance between the centers is `sqrt{(8- -4)^2+(5- -2)^2}=sqrt{193}approx 13.9` but the sum of the radii is `sqrt{9}+sqrt{25}=8` so these aren't close enough to touch, the closest points being along the line between the radii, `sqrt{193}-8` units apart.

Explanation:

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