Parabolas and hyperbolas

We’re now going to study two more conic sections. Two other types of curves that enter into– for example– the orbit that a comet can have, and things like that. Let’s begin with the parabola. Classically, a parabola begins by specifying in the plane– a point, F– and also a straight line– called the directrix. So here they are. As you know, when you have a point, P, you can define the distance, d, from the point P to the point F. But you can also talk about the distance from the point P to the straight line that I’ve given. And that’s going to be d as well. So to summarize– a parabola is going to be specified by the following thing.

Having fixed F, and the directrix– the straight line– I’m looking at all points, P, whose distance to F is the same as the distance to the line.

That’s the definition of a parabola– geometrical definition. As usual our main theme here, is to find a more concrete, more numerical, more equational way to describe parabolas. How do we do that? Well, we introduce coordinate axes. Now here I’ve introduced them according to the canonical configuration. What that means is the x-axis is going to be parallel to my initial given line. I’m also going to place the origin in such a way– in this canonical configuration– that it’s halfway between the point F and the initial given line. Therefore I’m going to call this distance between F and the line– 2p. So p is the distance between F and the origin. And it’s also the distance from the origin to the line.

So therefore the point V– which, here, is the origin– is clearly a point on my parabola, because it satisfies the condition that its distance to F is the same as its distance to the directrix. When you fill in the points that you get on a parabola, you get something that has a shape like this.

It turns out that the canonical equation of a vertical parabola– that is, when you describe in terms of equations what the distance condition means– gives you the equation y equals x squared over 4p. And this is called the canonical form of the equation of a parabola. The p is a positive parameter here. There is some jargon that goes with parabolas, just like there was a certain amount of terminology with ellipses. We defined the vertex to be the lowest point of our parabola. Here it’s the origin– 0, 0– in our canonical configuration. And the focus of the parabola is to point 0, p– the point F. The focus has a very interesting geometrical property, well, several really.

But one of them is even related to the use of satellite signals, for example, also important in optics. Let me describe it to you. If you have a straight line coming in parallel to the y-axis– and it bounces on your parabola by the usual law of incidence– that straight line will then proceed to go through the focus, F. This is true for any vertically approaching line, such as the following. Now if you imagine, for example, that your straight lines are incoming signals from a satellite, why would they be parallel? Well, because the satellite is so far away, that essentially its signal lines– when they reach your satellite dish– are parallel.

All those signal lines will reflect on the dish in such a way that they go through the F– the focus– of the parabola. In other words, they concentrate the signal at the focal point. This is an extremely important property, which explains why your dishes have a parabolic cross-section.

Now, we’ll look at curves that have other interpretations– namely hyperbolas. Now a hyperbola starts out a bit like an ellipse. That is, I give you– in the plane– two points F1, and F2. But the condition defining the hyperbola is going to be different. Let’s call the distance between the two points– as before– 2c. c is a positive parameter. The canonical configuration for our hyperbolas will be the case in which the two points are on the x-axis, and the origin is at the midpoint of the segment, F1 F2. And this is the canonical configuration. And we’d like to know the equation of a hyperbola, which I haven’t defined yet. Here’s how we define it.

Typical point P will have a distance– d1– to the first point–F1– that’s given, and d2 to the second point. We’re going to require– of our points on the hyperbola– that they satisfy the following condition. For a given parameter a– strictly between 0 and c– we want the difference between these two distances, d1 and d2, to be 2a– in absolute value. In other words, the difference between the two distances is plus or minus 2a. What does that describe? Well, when you work it out you get a curve that looks like this. The reason the curve now has two branches– a left branch and a right branch– is because the difference of the distances could be either plus 2a or minus 2a.

So it gives you two cases– two branches. The two points where the x-axis is crossed, of course, turn out to be the points a, 0, and minus a, 0. The distance, c– the original parameter– simply corresponds to the point F1, which is the point, then, c, 0. And if we define now, another parameter, b, by means of the equation b squared equals c squared minus a squared– then we can write the canonical equation of a hyperbola in the following form. It’s the set of points x, y– in the plane– satisfying the equation x squared over a squared, minus y squared over b squared, equals 1.

So you see the equation looks exactly like the equation of an ellipse– except there’s a minus sign, where before there was a plus.

That’s the canonical equation. And there is a certain amount of jargon for hyperbolas. There’s one focus, two focus, two foci. They are the points c, 0 and minus c, 0– the points F1, F2. The vertices, then, are the points where the hyperbolas cross the x-axis– the points plus or minus a, 0. And there are also something called asymptotes associated with a hyperbola. What’s an asymptote? Well, it’s a line that describes the limiting shape, or position, of the hyperbola. Here’s one of them. The line y equals b over a times x. It tells you what the hyperbolas are asymptotically approaching to– on the upper right, for example. And the negative of that line completes the picture in the other directions.

Now we’ve met the 3 conic sections– ellipses, parabolas, hyperbolas. We will see in the next segment why they are called conic sections.