Answer
The polar equation of $x+5y=8$ that expresses $r$ in terms of $\theta $ is $r=\frac{8}{\left( \cos \theta +5\sin \theta \right)}$.
Work Step by Step
The relation between polar coordinates and rectangular coordinates can be represented as below:
$x=r\cos \theta $ and $y=r\sin \theta $ ……(2)
Substitute values of $x\ \text{ and }\ y$ from (2) in (1) to get
$\begin{align}
& x+5y=8 \\
& r\cos \theta +5r\sin \theta =8 \\
& r\left( \cos \theta +5\sin \theta \right)=8 \\
& r=\frac{8}{\left( \cos \theta +5\sin \theta \right)}
\end{align}$
Therefore, the polar equation of $x+5y=8$ that expresses $r$ in terms of $\theta $ is $r=\frac{8}{\left( \cos \theta +5\sin \theta \right)}$.
