The equations represent a diagram of three tangent circles in a classic Sangaku : Japanese temple geometric puzzle . Background and solution can be found here : http://www.mfdabbs.pwp.blueyonder.co.uk/Maths_Pages/SketchPad_Files/Japanese_Temple_Geometry_Problems/Japanese_Temple_Geometry.html http://mathworld.wolfram.com/TangentCircles.html
falzoon2007-04-26T07:06:29Z
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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.