Show a polar equation describes a cylinder

Question

I have to show that $2r+r\cos\theta =1$ describes a cylinder.

I try moving the equation to cartesian coordinates and I get$\ 3x^2+4y^2-2x=1$, after that I don't know what to do, any help would be appreciated.

Answer 1

Note that $$1= 3x^2+4y^2-2x=3\left(x-\dfrac{1}{3}\right)^2+4y^2-\dfrac{1}{3}\Rightarrow \frac{\left(x-\dfrac{1}{3}\right)^2}{4}+\frac{y^2}{3}=1.$$

Then, you have an elliptic cylinder.