12.6 Ellipses  (Page 2/2)

Here is a drawing of a horizontal ellipse.

There are three numbers in this drawing. a is the distance from the center to the far edge (right or left). If you double this, you get the horizontal length of the entire ellipse—this length, $2a$ of course, is called the major axis.

b is the distance from the center to the top or bottom. If you double this, you get the vertical length of the entire ellipse—this length, $2b$ of course, is called the minor axis.

$c$ is the distance from the center to either focus.

$a$ and $b$ appear in our equation. $c$ does not. However, the three numbers bear the following relationship to each other: $a^2=b^2+c^2$ . Note also that a is always the biggest of the three!

A few more points. If the center is not at the origin—if it is at, say, ( $h$ , $k$ ), what do you think that does to our equation? They should all be able to guess that we replace $x^2$ with $(x-h{)}^2$ and $y^2$ with $(y-k{)}^2$ .

Second, a vertical ellipse looks the same, but with a and b reversed:

Let’s see how all that looks in an actual problem—walk through this on the blackboard to demonstrate. Suppose we want to graph this:

$x^2+9y^2–4x+54y+49=0$

First, how can we recognize it as an ellipse? Because it has both an $x^2$ and a $y^2$ term, and the coefficient are different. So we complete the square twice, sort of like we did with circles. But with circles, we always divided by the coefficient right away—this time, we pull it out from the $x$ and $y$ parts of the equation separately .

OK, get all that? Now, what do we have?

First of all, what is the center? That’s easy: (2,-3).

Now, here is a harder question: does it open vertically, or horizontally? That is, does this look like $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , or like $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ ? The way to tell is by remembering that $a$ is always bigger than $b$ . So in this case, the 36 must be $a^2$ and the 4 must be $b^2$ , so it is horizontal. We can see that the major axis ( $2a$ ) will be 12 long, stretching from (-4,-3) to (8,-3). (Draw all this as you’re doing it.) And the minor axis will be 4 long, stretching from (2,-5) to(2,-1).

And where are the foci? Since this is a horizontal ellipse, they are to the left and right of the center. By how much? By $c$ . What is $c$ ? Well, $a^2=b^2+c^2$ . So $c^2=32$ , and $c=$ $\sqrt{\text{32}}=4$ $\sqrt{2}$ , or somewhere around $5\frac{1}{2}$ . (They should be able to do that without a calculator: 32 is somewhere between 25 and 36, so $\sqrt{\text{32}}$ is around $5\frac{1}{2}$ .) So the foci are at more or less ( $-3\frac{1}{2}$ ,-3) and ( $7\frac{1}{2}$ ,-3). We’re done!

Homework:

There are two things here that may throw them for a loop.

One is the fractions in the denominator, and the necessity of (for instance) turning $25y^2$ into $\frac{y^2}{1/25}$ . You may have to explain very carefully why we do that (standard form allows for a number on the bottom but not on the top), and how we do that.

The other is number 7. Some students will quickly and carelessly assume that 94.5 is $a$ and 91.4 is $b$ . So you want to draw this very carefully on the board when going over the homework. Show them where the 94.5 and 91.4 are, and remind them of where $a$ , $b$ , and $c$ are. Get them to see from the drawing that $94.5+91.4$ is the major axis, and is therefore $2a$ . And that $a-91.4$ is $c$ , so we can find $c$ , and finally, we can use $a^2=b^2+c^2$ to find $b$ . This is a really hard problem, but it’s worth taking a lot of time on, because it really drives home the importance of visually being able to see an ellipse in your head, and knowing where $a$ , $b$ , and $c$ are in that picture.

Oh, yeah…number 6 may confuse some of them too. Remind them again to draw it first, and that they can plug in (0,0) and get a true equation.

OK, now we’re at a bit of a fork in the road. The next step is to connect the geometry of the ellipse, with the machinery. This is a really great problem, because it brings together a lot of ideas, including some of the work we did forever ago in radical equations. It is also really hard. So you can decide what to do based on how much time you have left, and how well you think they are following you. You may want to have them go through the exercise in class (expect to take a day). Or, you may want to make photocopies of the completely-worked-out version (which I have thoughtfully included here in the teacher’s guide), and have them look it over as a TAPPS exercise, or just ask them to look it over. Or you could skip this entirely, or make it an extra credit.