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.
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.
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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.
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: