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Families of Polar Curves: Conic Sections - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

One can use polar coordinates to describe polar conic sections, or conic sections with one focus at the origin of the polar coordinate system. Polar conic sections are described by a family of polar equations, and can be graphed using methods of graphing polar equations. Polar conic sections use similar methods for graphing as the rose or the cardioid.

Let's take a look at one final family of polar curves. r=a over 1 plus epsilon cosine theta or r=a over 1 plus epsilon sine theta. These are the conic sections with one focus at the polar origin. And let's graph one of these right now.
y=6 over 1 plus cosine theta. First I want to find if there is any symmetry with this graph. So I'm going to plug in negative theta and I get 6 over 1 plus cosine negative theta. Cosine's even so this will be the same as cosine theta 6 over 1 plus cosine theta and that's exactly r. So we know that r negative theta is in the graph and r negative theta is a reflection of r theta across the x axis. so this graph's going to be symmetric about the x axis. And we'll use that when we're graphing.
Okay. Let's plot some points. This is a mistake. This should be y r equals there we go. So, theta and r. Let's start with 0. When theta equals 0 cosine of 0 was 1, so it's 6 over 1 plus 1, 6 over 2 which is 3. And I'll skip to pi over 3. Cosine of pi over 3 is easier, it's one half. So this is 6 over 1 plus a half. 6 over 3 halves, 6 over 1.5 which is 4. And then pi over 2. Cosine of pi over 2 is 0 so it's 6 over 1, 6. And then 2 pi over 3. Cosine of 2 pi over 3 is negative one half. So it's 6 over 1 minus a half. 6 over 0.5, which is 12.
Now, notice what happens when cosine approaches pi or when that theta approaches pi. Cosine of pi is -1. So at pi this is going to be undefined. But imagine what happens as cosine gets close to -1. Say when it's a little bit short this will be a small positive number and so we get 6 over a small positive number. The r value is going to go to infinity. So what actually happens is that as theta goes to pi r goes to infinity.
Let's plot what we have so far and see what that looks like. We have 3 0, that's right here. Right. I have 6, 12 and 18 on this on this graph and we have 4 pi over 3. Pi over 3 is this direction so I go 4 that's here. Then 6 pi over 2. Pi over 2 is this direction I go 6 that takes me to here and finally I have 12 2 pi over 3. 2 pi over 3 is this direction, this is 6, 8, 10, 12. So the graph looks something like this. And remember it's symmetric across the x axis so I can I can reflect these points down. This one goes to here and this point goes here. And so just continuing around. We have a parabola. Remember that these conic sections all have a focus at the origin. So this conic section has a vertex at 3 0. It's got this distance is 6, this distance is 6. This length the width that passes through the focus, it's called the Latus Rectum. It's 12, exactly twice this value.
And so notice the epsilon value here is 1. Whenever the epsilon value is 1 or -1, you'll get a parabola. And this will always behalf of the Latus Rectum, the width of of the graph.

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