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If e 1 the graph is a hyperbola whose transverse axis

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Unformatted text preview: ) represent the pole regardless of the value of θ. The key to understanding this result, and indeed the whole polar coordinate system, is to keep in mind that (r, θ) means (directed distance from pole, angle of rotation). If r = 0, then no matter how much rotation is performed, the point never leaves the pole. Thus (0, θ) is the pole for all values of θ. Now let’s assume that neither r nor r is zero. If (r, θ) and (r , θ ) determine the same 11.4 Polar Coordinates 787 point P then the (non-zero) distance from P to the pole in each case must be the same. Since this distance is controlled by the first coordinate, we have that either r = r or r = −r. If r = r, then when plotting (r, θ) and (r , θ ), the angles θ and θ have the same initial side. Hence, if (r, θ) and (r , θ ) determine the same point, we must have that θ is coterminal with θ. We know that this means θ = θ + 2πk for some integer k , as required. If, on the other hand, r = −r, then when plotting (r, θ) and (r , θ ), the initial side of θ i...
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