Chapter 10 - Section 10.3 - Polar Coordinates - 10.3 Exercises - Page 667: 23

$r=\frac{1}{sin\theta-3cos\theta}$

$y=1+3x$ We know that: $sin\theta=y/r$, so $y=rsin\theta$ We also know that: $cos\theta=x/r$, so $x=rcos\theta$ Substituting for $x$ and $y$, we get: $rsin\theta=1+3rcos\theta$ $rsin\theta-3rcos\theta=1$ $r(sin\theta-3cos\theta)=1$ $r=\frac{1}{sin\theta-3cos\theta}$