# Chapter 11: Parametric Equations and Polar Coordinates - Section 11.3 - Polar Coordinates - Exercises 11.3 - Page 663: 68

a) $r=a \sec \theta$ b) $r=b \csc \theta$

#### Work Step by Step

a) Conversion of polar coordinates and Cartesian coordinates are as follows: 1. $r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$ 2. $\tan \theta =\dfrac{y}{x}$ or, $\theta =\tan^{-1} (\dfrac{y}{x})$ 3. $x=r \cos \theta$ and 4.$y=r \sin \theta$ Every vertical line in the xy-plane has the form of $x=a$ This means that $r \cos \theta=a$ Thus, the polar equation can be rearranged as: $r=a \sec \theta$ b) Since, we have $x=r \cos \theta$ and $y=r \sin \theta$ and $r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$ Every horizontal line in the $xy$-plane has the form of $y=b$ This means that $r \sin \theta=b$ Thus, the polar equation can be rearranged as: $r=b \csc \theta$

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