Answer
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: $r=b \csc \theta$
