Solution to GL’s ‘Sangaku’

GL decided to set us an optional prep in the form of a past A-level question. I now see that the adults who sat these papers last time really have a very good point about dumbing down – the difficulty of some of these questions is completely off the scale compared to anything any of us have done before in routine syllabus work (i.e. not BMO etc).

Anyways the thing was on parabolas and I’ve attached my solution for anyone who particularly wants to see it. The reason it’s called ‘solution to sangaku’ is because GL mentioned after drawing it on the board that it looked like it belonged to a genre of Japanese circle puzzles called sangakus which were apparently put on display in temples or some such public area for the amusement (and frustration) of anyone willing to sacrifice an hour or two to solving problems.

Linky for download of solution [docx]

The original question was something like:
Given a parabola with parametric equation [where the focus of the parabola is (a, 0)]:

x = at^2
y = 2at

and circles drawn as shown in the diagram I drew, prove that R – r = 4a. We were also given the equation of normals to the parabola at a point specified by t:

tx + y = at^3 + 2at

GL did mention as he was elucidating the inner workings of this subset of conic sections that the results and proofs and general maths tend often to end up rather beautiful. The solution to this particular one just seems to require a bit of luck (i.e. somehow having the instinct to take the difference of both formulae) and a whole load of cancelling at the end which made me want to shout something like “W00T” before realising any such exhortation would be inappropriate and probably unwelcome at 2 in the morning… Anyways the solution speaks for itself really – it’s especially awesome if you’ve had a go at the question. Enjoy ;)

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One Response to Solution to GL’s ‘Sangaku’

  1. Will says:

    Thanks – a welcome distraction from inorganic chemistry!

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