Writing the line $y=2x+1$ in the polar form

Question

By letting $x=r\cos(\theta)$ and $y=r\sin(\theta)$, I have obtained $r=\dfrac{1}{\sin(\theta)-2\cos(\theta)}$.

Can this polar equation actually represent the line?
I am curious because the denominator is undefined for some values.

Thank you.

Answer 1

I would say the better representation is the equation $$r(\sin\theta - 2\cos\theta)=1$$ where you avoid the undefined cases when $\sin\theta -2\cos\theta = 0$. In this equation, it's clear that such points are not on the line (since the left side of the equation is $0$). In fact, those points are on the line $y=2x\color{red}{+0}$