A mystery - The 4U rectangular hyperbola rotation proof

[RuiAce]

Hello all,

So, I was casually at UNSW procrastinating on a lot of important stuff, and I decided to ask myself this question.

This is something that's often overlooked and just taken for granted, because it only appears 1% of the time in the 4U course. But it's quite important though, because we define the rectangular hyperbola from our usual hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) to be the case when \(b=a\). We can do a whole bunch of nice proofs using just this, but we insist on using the second version anyway because the equation is nicer. Therefore, the link between the two should be justified.

For a long time, I thought "hey, I should go prove it", but then I'd forget about it a few minutes later (because I was either busy or just being lazy). I was finally bothered enough to find a proof today, so I'll present it here. A note that I'm fairly sure this proof has been reproduced hundreds of times in the past (by the time you're going into 3rd year uni math, this should be easy) but I haven't found it on the internet anywhere, so I'll present it here.

At least, I haven't found a 4U adequately-friendly proof. I personally think this proof is easier when I use linear algebra, but in 4U the best I can deal with is complex numbers.
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The basic idea is that whereas \(z\) is written in terms of \(x\) and \(y\), \(w\) is written in terms of \(u\) and \(v\) instead. Since \(w\) is no longer written in terms of \(x\) and \(y\), we have to adjust our final result accordingly, and hence it includes \(u\) and \(v\) as well.
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