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: 23


The answers: (a) $r=2$ ${\ \ \ }$ (iii) Circle (b) $\theta=2$ ${\ \ \ }$ (iv) Line through origin (c) $r = 2\sec \theta $ ${\ \ \ }$ (i) Vertical line (d) $r = 2\csc \theta $ ${\ \ \ }$ (ii) Horizontal line

Work Step by Step

(a) When $r=2$ , the distance of the curve is $2$ from the origin, so it is a circle of radius $2$. The answer is (iii) Circle. (b) When $\theta=2$, it is a line through the origin that makes an angle $\theta=2$ with the $x$-axis. So, the answer is (iv) Line through origin. (c) When $r = 2\sec \theta $, we have $2 = r\cos \theta = x$. The $x$-coordinate is $2$, a constant. So, it is a vertical line at $x=2$. The answer is (i) Vertical line. (d) When $r = 2\csc \theta $, we have $2 = r\sin \theta = y$. The $y$-coordinate is $2$, a constant. So, it is a horizontal line at $y=2$. The answer is (ii) Horizontal line.
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