Consider the following equation in polar coordinates, where r0 is a positive constant

Chapter 14, Problem 14.3.21

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Consider the following equation in polar coordinates, where r0 is a positive constant and where 0 < 1 is a constant: r = r0 1 cos , 0 < 2. (a) Show, by converting to Cartesian coordinates, that this equation describes an ellipse. You may use the results of 20. (b) What are the minimum and maximum values of r? At what values of do they occur? (c) Graph the ellipse for r0 = 6 and = 0.5, labeling the points from part (b) as well as the y-intercepts, that is, the points at = /2 and = 3/2. What is its center? (d) Find a formula in terms of r0 and for the length of the horizontal axis of the ellipse. (e) The constant is known as the eccentricity of the ellipse. Describe in words the appearance of an ellipse of eccentricity = 0. What happens to the appearance of the ellipse as the eccentricity gets closer and closer to = 1?

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