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Families of Polar Curves: Conic Sections - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Let's look at some polar equations, and ask the question; what would the graphs look like? Starting with this one; r equals 2 over 1 minus cosine theta.
Now we've been studying this family of functions, r equals a over 1 plus epsilon cosine theta. This is really close to the form we have here, but not quite. There's a minus sign rather than a plus sign. However, we can use an identity, this identity; cosine of theta plus pi, equals minus cosine theta, to rewrite this in this form. So I can replace the negative cosine theta with a plus, cosine of theta plus pi. So this equals 2 over 1 plus cosine theta plus pi.
So when we look at this, we can see that epsilon is 1. From our experience, if epsilon is 1 means that this graph is going to be a parabola. What effect does this transformation have on the graph? Now if we were graphing a rectangular, theta plus pi inside the function, this would indicate some kind of shift to the left, shift in the negative direction pi units. But we're in polar and theta is an angle. So when we add pi, this is going to be a rotation in the negative direction pi.
So we're rotating. Let's say the graph of the original parabola looks like this. We're rotating this graph 180 degrees clockwise, that's the negative direction. So it's going to end up looking like this. So this is a parabola rotated pi clockwise. So that's what this represents. So the effect of changing the plus to a minus is actually going to reverse the graph.
Now let's take a look at another example. R equals 5 over 1 plus 2/3 cosine of theta minus pi over 4. Now this 2/3, this is our epsilon, and when epsilons between 0 and 1, we know that this graph represents an ellipse. Now, what else can we tell from this equation? Well, we have some kind of rotation here, just like we did in the previous example. Except, this is a rotation in the positive direction, pi over 4, so an ellipse rotated pi over 4, and the positive direction is counter clockwise.
Well, let's think about what that means. An un-rotated cosine ellipse is going to be symmetric about the x axis. If we rotate the graph pi over 4 counter-clockwise, that axis of symmetry rotates as well. So if the original graph looks like this, the new graph is going to look like this. So the axis of symmetry will now be theta equals pi over 4. So the effect of adding or subtracting some number to theta gives us a rotation.
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