A geometric puzzle for your sphere time ...?

The equation to a circle is :

(x - h)^2 + (y - k)^2 = r^2

where (h,k) is the centre and r is the radius.

Let's complete the squares :

(1) x^2 + y^2 - 18y = 0

or, x^2 + y^2 - 18y + 81 = 81

Thus, (x - 0)^2 + (y - 9)^2 = 9^2

This is a circle with radius = 9, centred on (0, 9).

(2) x^2 - 24x + y^2 - 8y + 144 = 0

or, x^2 - 24x + 144 + y^2 - 8y + 16 = 16

Thus, (x - 12)^2 + (y - 4)^2 = 4^2

This is a circle with radius = 4, centred on (12, 4).

(3) x^2 - 14.4x + y^2 - 2.88y + 51.84 = 0

or, x^2 - 14.4x + 51.84 + y^2 - 2.88y + 1.44^2 = 1.44^2

Thus, (x - 7.2)^2 + (y - 1.44)^2 = 1.44^2

This is a circle with radius = 1.44, centred on (7.2, 1.44).

The radii are 9, 4 and 1.44, and you want ME to figure out

how they are related?? I cannot find a simple relation there.

Perhaps it has something to do with the word "sphere",

i.e. 3-dimensional??

What about, if we take 9, 4 and 1.44 and multiply them each

by 25. Then we get 225, 100 and 36.

Now we see that :

225 - 100 = 125 = 5^3

and

100 - 36 = 64 = 4^3.

That looks like a nice relationship!

Taking 225, 100 and 36 again, these are 15^2, 10^2 and 6^2

and 15, 10 and 6 are all triangular numbers.

As you can see, I'm guessing wildly here. I really have no

idea of what's required, but it was an interesting exercise.