Polar Equations

By Courtney Cody

In this assignment, we will investigate the following polar equations for different values of p:

for k > 1, k = 1, and k<1.

LetÕs begin by considering just the first polar equation, , for different values of p and k.

First weÕll consider the general case of this polar equation p = k = 1:

By studying the graph, we first notice that our equation for r is not a function since it does not pass the vertical line test.  It does, however, look like a sideways parabola opening up to the right.  To confirm this observation, letÕs observe the graph when k = 1 and p = 4.

Once again, we see that the graph when k = 1 and p is varied is a parabola, only this second graph is wider.  Hence it appears that p is a scalar and that the k = 1 causes the graph to be a parabola.  Thus, the equation, , where k = 1 and p is varied is the polar equation of a parabola.

Next weÕll consider the case of this same polar equation k < 1 and p is varied.  Consider the following graphs for k = 0.5, p = 1; k = 0.5, p = ; and k = 0.5, p = -2, respectively.

By observing the graphs when k < 1 and p is varied we see that it is an ellipse.  Another aspect of the graph that we can see is the furthest distance a point on the ellipse is away from the origin is p units.  Additionally the y-intercepts are always .

Once again weÕll consider this same polar equation but for the case when k > 1 and p is varied.  Consider the following graphs for k = 1.1, p = 1; k = 1.1, p = ; and k = 1.1, p = -2, respectively.

By observing the graphs when k >1 and p is varied we see that it is a hyperbola and its asymptotes.

Now we will consider the second polar equation, , for different values of p and k.

First weÕll consider the general case of this polar equation p = k = 1:

By studying the graph, we notice that it looks like a sideways parabola opening up to the left.  To confirm this observation, letÕs observe the graph when k = 1 and p = 4.

Once again, we see that the graph when k = 1 and p is varied is a parabola, only this second graph is wider.  Hence it appears that p is a scalar and that the k = 1 causes the graph to be a parabola.  Thus, the equation, , where k = 1 and p is varied is the polar equation of a parabola.

Next weÕll consider the case when k < 1 and p is varied for the second polar equation.  Consider the following graphs for k = 0.5, p = 1; k = 0.5, p = ; and k = 0.5, p = -2, respectively.

By observing the graphs when k < 1 and p is varied we see that it is an ellipse similar to the one for the first polar equation.  In addition, we can see the furthest distance a point on the ellipse is away from the origin is once again p units and the y-intercepts are always .

Seeing as how the conic sections have been of the same type for the different polar equations thus far, what do we expect for the case when k > 1 and p is varied?  For the previous polar equation we graphed a hyperbola.  To test our hypothesis, weÕll consider the following graphs for k = 1.1, p = 1; k = 1.1, p = ; and k = 1.1, p = -2, respectively.

By observing the graphs when k >1 and p is varied we see that it is a hyperbola and its asymptotes, as we expected!

For the next two polar equations, we will investigate the effect on the graphs from replacing  with .  We will begin by considering the second polar equation, , for different values of p and k.

First weÕll consider the general case of this polar equation p = k = 1:

By studying the graph, we notice that it looks like a parabola opening up.  This is the same parabola as in the first polar equation only rotated 180 degrees.  To confirm this observation, letÕs observe the graph when k = 1 and p = 4.

Once again, we see that the graph when k = 1 and p is varied is a parabola, only this second graph is wider.  Hence it appears that p is a scalar and that the k = 1 causes the graph to be a parabola.  Thus, the equation, , where k = 1 and p is varied is also a polar equation of a parabola.

Next weÕll consider the case when k < 1 and p is varied for the third polar equation.  Consider the following graphs for k = 0.5, p = 1; k = 0.5, p = ; and k = 0.5, p = -2, respectively.

By observing the graphs when k < 1 and p is varied we see that it is an ellipse that is elongated vertically.  In addition, we can see the furthest distance a point on the ellipse is away from the origin is once again p units.  In this case, the x-intercepts are always .

Seeing as how the conic sections have been of the same type for the different polar equations thus far, what do we expect for the case when k > 1 and p is varied?  For the previous polar equations we graphed a hyperbola opening to the left and right.  Since our ellipse rotated 180 degrees when we replaced  with , we expect our hyperbola to rotate similarly.  To test our hypothesis, weÕll consider the following graphs for k = 1.1, p = 1; k = 1.1, p = ; and k = 1.1, p = -2, respectively.

By observing the graphs when k >1 and p is varied we see that it is a hyperbola opening upward and downward and its asymptotes, as we expected!

For the fourth polar equation, we will consider the second polar equation, , for different values of p and k.

First weÕll consider the general case of this polar equation p = k = 1:

By studying the graph, we notice that it looks like a parabola opening downward.  This is the same parabola as in the second polar equation only rotated 180 degrees.  To confirm this observation, letÕs observe the graph when k = 1 and p = 4.

Once again, we see that the graph when k = 1 and p is varied is a parabola, only this second graph is wider.  Hence it appears that p is a scalar and that the k = 1 causes the graph to be a parabola.  Thus, the equation, , where k = 1 and p is varied is also the polar equation of a parabola opening downward.

Next weÕll consider the case when k < 1 and p is varied for the fourth polar equation.  Consider the following graphs for k = 0.5, p = 1; k = 0.5, p = ; and k = 0.5, p = -2, respectively.

By observing the graphs when k < 1 and p is varied we see that it is an ellipse that is elongated vertically.  In addition, we can see the furthest distance a point on the ellipse is away from the origin is once again p units.  In this case, the x-intercepts are always .

Using the patterns we have observed for the different polar equations thus far, what do we expect for the case when k > 1 and p is varied for the fourth polar equation?  For the previous polar equations we graphed a hyperbola opening to the left and right.  Since our ellipse rotated 180 degrees when we replaced  with , we expect our hyperbola to rotate similarly.  To test our hypothesis, weÕll consider the following graphs for k = 1.1, p = 1; k = 1.1, p = ; and k = 1.1, p = -2, respectively.

By observing the graphs when k >1 and p is varied we see that it is a hyperbola opening upward and downward and its asymptotes, as we expected!

Return