So sure enough, the 153 student… “Charles”, say (since that’s nobody’s name in the class)… that stayed after class last time to talk about the parabola problem had nailed it; it’s pretty doggone sweet so here goes.
Consider a parabola in the plane: the “locus” (a term with, for me, a sort of archaic feel) of points, P, such that the distance from P to a certain fixed point (the focus, F) is equal to its distance from a certain fixed line (not passing through F; the directrix [d, I suppose… though I doubt I’ll need to refer to it in code]). Dropping a co-ordinate frame (with the usual (x,y) notations) having its origin at F and its x-axis parallel to the directrix, scale so that the vertex is at (0, -1); the equation of the parabola is then 4(y+1)=x^2 (by Math 150—the “generic” 4A(y-K) =(x-H)^2 has [directed, vertical] distance A between vertex and focus, and vertex (H,K)).
Now sketch the tangent line to the parabola at P. We aim to show that the angle formed by this line and the line through points F and P is equal to the angle formed by the same tangent line and a vertical line. (When vertical lines are considered as “sunlight”, this “reflecting property of the parabola” will cause the reflected beams to concentrate at the focus [and fry the bug we’ve sadistically put there… hsss]).
Okay. Verticals are at angle (i.e., at 90 degrees—as is usual in calculus we’ll use radian measure; also as usual angles are measured from the positive x-axis [“standard position”])… so we get that for free. Following up on an example of the text, we can “parameterize” the parabola using the slope, M; since y’ = x/2 one has x = 2M; it follows that y = M^2 – 1.
Now, I’m not even sure myself at this point if this was a particularly useful thing to do… it may turn out that it’d’ve been easier not to parameterize… but the point of even trying this was that one also has a concise representation of another number we’ll need for our calculations: the “standard position” angle of a line with slope M is simply .
The other line in question, passing through P and the focus, has slope —this is why we put the origin at the focus. Referring to our figure (alas left to the reader; computer graphics is hard) we see that we require
Finally the magic: put … so we’re looking at
Apply the tangent function on both sides:
But this is bygod just a familiar trig identity… namely, the “tangent double-angle law”
. Take that.
Because, while this isn’t quite the proof we should probably seek—that proof would involve careful checking that going “backward” from the identity we’ve ended with to the equation we seek (a certain pair of angles) won’t result in any snafus—nevertheless I’m convinced and I’m quitting here.
Because the real reason for the post is for me to say that Charles’s contribution in reaching this result was substantial: there were at least a couple of points where he was the first to arrive at a necessary insight.
Now. How much of this kind of really-doing-math can I reasonably expect to get with, well, say an A or a B or a C student? Ideally, quite a bit. And somewhere along the line I can imagine figuring out how. As of now I’m openly kind of flailing about. Which has to be OK. Because it’s the only way I know how to learn anything and in particular it’s essentially what I’m trying to get students to do. There’s a lot of groping around in the dark when you’re mathing.
I’d hate to tell you how long it took me to chase down the right equation for that parabola during this writeup, for example. I will go so far as to tell you that I first noticed something wrong at the double-angle law. (Charles and I had it right the first time, you understand. But one remembers ideas not formulas mostly and so I lost… oh, half-an-hour easy… finding a simple mistake. This kind of thing happens all the time and is presumably just the kind of thing that accounts for the tremendous mass appeal of math as a subject of study.)
So I flail about figuring out how to get ’em to willingly do math with me and their peers… and they flail about with, sometimes, stuff I persist in calling “routine exercises” or even, (but always with scarequotes) ” ‘easy’ problems”. And big-picture stuff… and everything in between.
Mathematics begins in confusion… and ends in confusion. I wish I knew who said that. Anyhow, it’s true. Once you understand something, you’ve done some math: so remember where you put it in case you need it again… and find something else to think about… something you don’t understand…
Or type it up and post it on the net.