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Unformatted text preview: Intersections of Polar Curves The purpose of this supplement is to find a method for determining where graphs of polar equations intersect each other. Let’s start with a fairly straightforward example. Example 1. The graphs of r = 1 and r = 2 cos θ are sketched in the figure below. Where do they intersect each other? It appears that they intersect twice, once in the first quadrant and once in the fourth. One way that we might try and determine the intersection is by solving the following system of equations for r and θ . r = 1 r = 2 cos θ We first set these equal to each other and solve for θ , as follows: 2 cos θ = 1 cos θ = 1 / 2 θ = π/ 3 , 5 π/ 3 We only list possible solutions between and 2 π because by the time θ has run from to 2 π each graph has been traversed at least once. 2 Now we use the equation r = 1 to find the r-coordinates of the points we are interested in. The intersections must occur at the points (1 , π/ 3) and (1 , 5 π/ 3) . Essentially what we did in the previous example was to solve the equations simul- taneously (in other words, we thought of them as a system of equations and solved that system). This is exactly what we do with rectangular equations, so there doesn’t really seem to be anything new here. Unfortunately, this method doesn’t always work. Let’s look at another example. Example 2. Find the intersection of the graphs of r = cos θ and r = 1 − cos θ . The graphs are sketched below. There appear to be three intersection points to look for. Let’s first try to do what we did in the previous example. Our system of equations is: r = cos θ r = 1 − cos θ These are already solved for r , so we set these expressions equal to each other and solve: cos θ = 1 − cos θ 2 cos θ = 1 cos θ = 1 / 2 θ = π/ 3 , 5 π/ 3 Now plug these values for θ into the first equation to find the intersection points...
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- Spring '09