SWBAT graph conic sections that are in polar form.

All conic sections have a directrix? Where have these been?

10 minutes

The purpose of today's lesson is to think about how we can write the **equations for conic sections in polar form** instead of the rectangular form that they are used to. One change I establish right away is that the focus of the conic section will be located at the polar origin instead of the center.

**Eccentricity is an important part of this lesson** so I give students this notes worksheet and have them work on #1-3 with their table group. After a few minutes we share out to make sure that they recall this key concept. I also want them to realize that eccentricity is between 0 and 1 for an ellipse, equal to 1 for a parabola, and greater than 1 for a hyperbola. This will help us identify the type of conic section when it is written in polar form.

25 minutes

**Teacher Note:** Here is a teacher copy of the notes so you can see what my intents are for this exploration. This will be a whole-class discussion so that we can efficiently derive the polar form of a conic section.

After we review eccentricity, I explain to students that a parabola is not the only conic section that has a directrix and that all of them can be defined in this way. On page 2 and 3 I **sketch the focus and directrix of each conic section** and explain that PF/PQ in each case will equal the eccentricity of the conic.

Then, we can use that fact to **derive a general formula **for an ellipse in polar form. I outline the process in the video below.

After this, I explain that this general formula might change depending on whether the conic is horizontal or vertical and where the directrix is located. Then I show the different forms on page 4 and explain what each formula represents.

10 minutes

After we come up with this general form, I will have students give #1 and #2 on the last page a try as their homework. Before they get started, I will ask them how it looks different than the formula we just derived and they usually notice that there is a 3 in the denominator instead of a 1. Once they realize this I ask a student what we should do and they suggest dividing everything by 3 to make it a 1.

At this point I will turn them loose and let the students work on these problems for the rest of class. Here are **some hints I give if students get stuck**:

- If they are putting the center at the origin, I will remind them that for the polar equation the focus is at the origin.
- If they don't know what type of conic it is, I will ask what the eccentricity is and see if that will help them determine the type.
- If they don't know how to graph it, I will suggest that they plug in very easy points in order to get a rough sketch of the conic.