Curve shape prediction by changing configuration space (Cartesian to polar)

Question

Let us consider the equation $y=3x+2$ which describes a straight line in the 2D Cartesian space. Is it possible to predict the shape of this curve in the polar ($r,\theta$) space? How?

I believe that the transformation from Cartesian to polar coordinate system imposes that :

$r=\sqrt{x^2 + (3x+2)^2}$

$\theta=\arctan\left(\frac{3x+2}{x}\right)$

How can this help predict the shape of the $y$ curve in the polar ($r,\theta$) space?

Answer 1

Using $x=rcos\theta$ and $y=rsin\theta$ we get $rsin\theta=3rcos\theta+2$ which is $r(sin\theta-3cos\theta)=2$ Dividing gives $r=\frac{2}{sin\theta-3cos\theta}$ This is the transformation from Cartesian to Polar describing the same line