So we’ve had a look at linear and quadratic relations. Now we’re going to focus on inverse relations, which are a little bit more complicated, but very, very interesting and found in many different places. So the basic geometry there goes back to Apollonius, to the ancient Greeks, but connects naturally to this famous problem of the motion of the planets and Newton’s resolution of Kepler’s equations. Inverse relationships occur when one quantity times another quantity gives a fixed constant– say x times y equals a. In that case, if you double x, then you half y. If you triple x, then y is reduced by a third.
So this balancing situation that happens in an inverse relation– one goes up and the other goes down– is quite different to the linear and quadratic relations that we’ve talked about previously. So we’ve seen that conic sections go back to the ancient Greeks in terms of slicing a cone with various planes. And we talked about ellipse, parabola, and hyperbola. These are the three basic conic sections studied by Apollonius. Apollonius, however, also knew that there were other ways of thinking about these conic sections– in particular, there’s an important metrical way of distinguishing between them. And that involves a fixed point called a focus and a fixed line l, called the directrix.
So it turns out that we’ve already seen that the parabola can be described as the locus of a point x, which is always equidistant from the focus to the line. So this segment and the segment are always equal. That defines a parabola. In a similar fashion, if we modify that just a little bit and replace this 1 here with a number less than 1– say, 1/2– so we look at points x so that the distance from f to x divided by the distance from x to the line is, say, always 1/2. Then we trace out this ellipse.
And today, we’re going to be very interested in this other case where this constant here, called the eccentricity, is actually a number bigger than 1. So the distance from f to x divided by the distance from x to the directrix l is, in this case, 2. And that gives us a curve that has two branches, like this. This is a hyperbola, and it has a number of interesting features. Prominent amongst them is the existence of certain special lines. So it turns out that if we zoomed away from the hyperbola, then we would find that it looked very much like a pair of lines.
So as we go further out, the branches of the hyperbola approach the two pairs of lines, called the asymptotes. So this is a very interesting classical curve studied by Apollonius, and we’re going to see that it’s really intimately connected with the idea of an inverse relation. In the 17th century, mathematicians discovered a solution to this famous problem of what actually happens up in the night sky. This is an issue that had fascinated peoples for millennia. And due to the combined efforts of Copernicus, Tycho Braye, Kepler, and then ultimately Newton, the whole situation was pretty well nailed down.
So what actually happens is that planets go around the sun, not anything else, and they go in elliptical orbits with the sun as one of the foci. So generally, an ellipse has, in fact, two foci, but the sun is always at one of them. And Kepler’s laws also states something about how fast the things go. So basically, if we join a planet with the line to the sun, then this area that’s swept out is, say, constant in different time intervals. So if it’s closer to the sun, its moving faster, and where it’s further from the sun, it’s moving slower. That applies also to comets, which are returning, like Halley’s Comet.
They’re on elliptic paths with the sun at focus, and they’re rather slow out in this region. But when they get closer to the sun, they speed up and zip around it. So it’s important to realise how the geometry of conic sections was crucial for this. And this basic framework of using coordinates, of course, also an important ingredient. And where do hyperbolas fit into the situation? Well, every so often, a stray object– maybe from outer space– is going to come. It’s going to be coming close to the sun, in which case it’s going to be attracted to the sun and it’s going to be bend like this. And then its path ends up being a hyperbolic path.
Although in fact, actually, most of the observed comets of this kind have eccentricities very close to 1. So they’re, in fact, actually pretty close to being parabolas. So what about the geometry of a hyperbola looked at from the Cartesian point of view? Well, it turns out that it’s very beautifully captured by a very simple equation– y equals 1 over x. This equation– somewhat miraculously– give us exactly the same structure that Apollonius’s cutting of the cone does. Another way of representing the curve is through the equation xy equals 1, where we see this basic inverse relation property of two things multiplying to give a fixed constant. So let’s have a look at this curve. Very important for us.
If x is 1, then y is 1. If x is 2, y is 1/2. If x is 3, the y is 1/3, and so on. As x gets bigger and bigger, y approaches 0. So we’re approaching this asymptote, which is the x-axis, with the equation y equals 0. On the other hand, if x is approaching 0, then the y value gets very big positives. We’re approaching this vertical asymptote. It’s the y-axis given with equation x equals 0. Notice that the actual form of the function here does not allow x equals 0, because then the thing is undefined.
And notice that when we cross over that singularity, well, the y values have a very abrupt discontinuity and they become very large negative quantities.
So this interpretation here is very nice geometrically, because it’s telling us that if we choose any one of these points on this curve, then we look at the rectangle formed, then that rectangle always will have area 1. That’s basically what it’s saying. The x-coordinate times the y-coordinate is always equal to 1. So this red rectangle and the green one and the black one, they all have area 1, suggesting that there’s something rather interesting about the area underneath this hyperbola, and that turns out to be a very important and profound effect.
So this Cartesian framework is a very powerful tool to allow us to extract properties of the hyperbola to prove things and to apply it in a lot of different situations. Very nice, clean, algebraic form for a beautiful geometrical object.