College Algebra: Introduction to Hyperbolas

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Well, now it's time to talk about hyperbolas. Yes, these are so neat. So, what's a hyperbola? Well, you can think of it in terms of iconic section. By just taking iconic section and slicing like this and you get this hyperbola, which actually has two wings. It has a wing sort of on one side and then a wing on the opposite side. But how do you describe that thing analytically? Well, you can describe a hyperbola analytically, no problem at all.
Well you could think about it, if you must, as the following. You again have two foci. Like maybe one over here, and one over here for example. But now you're looking at all the points that have the property that if you take the difference, not the sum, but the difference between the length from that point to one focus minus the length from that point to the other focus. That difference is always constant. Remember, if the sum is always constant you get an ellipse. But if the difference is always constant then it turns out what you get is something that looks more like this. I'll try to draw this in for you. This is a hyperbola.
The general shape of a hyperbola that sort of looks like this is the following. It's ^x²/[a²] - ^y²/[b²] = 1. Where in fact notice that here this thing crosses the x-axis and so therefore it's the x thing and I'm subtracting the y thing. It's an important distinction here. If this thing is going to cross the x-axis it's going to cross it at a and minus a. You can see that. Because it crosses the x-axis when y is zero. If y were zero, notice I have ^x²/[a²] = 1. So, that means that x = a. So, that's right. But notice if I make x = 0 I can't solve this. Because 1 can never equal negative something squared. It can't be solved. So, in fact, it never crosses this axis.
So, this is how this thing looks. This is called sometimes the traverse axis. The traverse axis is right here. In fact, it turns out that a hyperbola actually has asymptotes. The asymptote sort of looks like this. These are lines that the hyperbola approaches but in fact never touches. You can give the equations for this pretty easily. The equation for the asymptotes will always be y = ^b/[ax]. So y = +^b/[ax] would look like this. y = -^b/[ax] might look like this. Those are two lines that the curve sort of approaches, but never actually touches.
Now, how do you find these foci? How do you find these points? Well, those are pretty easy too, because if I call this c and this would be minus c, then--oh, I'm sorry. That's actually not quite right, because the foci are actually way over here. This is actually wrong, don't look at that. Now, the foci are actually somewhere in here you see, somewhere inside this. Just like the parabola. I call that c and I call this minus c. You can actually find that value pretty easily. C is going to be equal to the following. If I take it and square it, it's going to be a² + b². So, very much like the ellipse. You can see that these things are almost cousins. There it was a² - b². Now, I see c² would equal a² + b². So, you wanted to find c, just take the square root of that.
So, you just take these two numbers here and add them and take the square root and that would give you c. Its negative will give you the other foci. So, not a hard thing at all to find the foci, if you just know this formula, just follows this thing. What if, in fact, the hyperbola was the other way? Well, then the y would come first and the analysis would be the exact same. Let me just show you that really fast. So, in that case, now the asymptotes would look something like this. I'll put in the asymptotes. What you have here now, the equation of this would look like this. Now, you'd have ^y²/[b²] being the first term and then I subtract off the ^x²/[a²] and that equals one. So, it would look something like that,
You see how now the y² is in the front and I'm subtracting the x²? It doesn't cross the x-axis. Now it crosses the y-axis and where? It would cross it at b and minus b. Same as before. Now, where would the foci be here? Well, the foci would be somewhere over here. If I call this c and this minus c, then I again have this c² = a² + b², but now the foci would be located at (0, c). Whereas here, in this example, the foci you could see are located at (c,0) because they're on the x-axis. Here they're on the y-axis and so I see (0, c). So, everything else remains the same and if you want to figure out the asymptotes. The asymptotes for this kind of thing would be y = ^b/[ax]. Just like before.. So, that doesn't change either.
So, the asymptote lines are going to be found the same way. You take always the b over the a, plus or minus, and those give you the asymptotes. So, you could find the foci, you could find the asymptotes, you could sort of put the whole thing in and you can get a hyperbola. Let's take a look at some hyperbola coming up next.
Conic Sections
Hyperbolas
An Introduction to Hyperbolas Page [1 of 1]