Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.3 Polar Coordinates - Exercises - Page 618: 27


$r = u\sec \left( {\theta - \frac{\pi }{3}} \right)$, where $u$ is any real number.

Work Step by Step

Since $r\cos \left( {\theta - \frac{\pi }{3}} \right) = 1$ is a line, the polar equation can be written as $r = \frac{1}{{\cos \left( {\theta - \frac{\pi }{3}} \right)}} = \sec \left( {\theta - \frac{\pi }{3}} \right)$. The slope of this line is given by $\tan \theta = \frac{y}{x} = \frac{{r\sin \theta }}{{r\cos \theta }}$. Notice that we can scale $r$ by multiplying it with any number such that the slope does not change. Thus, we obtain a family of lines parallel to the original line by multiplying $r$ with a constant. Hence, the polar equations of the lines parallel to the line $r = \sec \left( {\theta - \frac{\pi }{3}} \right)$ is just $r = u\sec \left( {\theta - \frac{\pi }{3}} \right)$, where $u$ is any real number.
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