How to think in terms of Polar Coordinates? I am currently studying Polar Coordinates and many times I've noticed that one converts polar equations in cartesian form to do further analysis. Is there a way to think in terms of Polar Coordinates? Answers to this question must include a way in which one can reason within the domain of polar coordinates. 

For starters, consider the following problem: 

Find the equation of the line perpendicular to $l\over{r}$=$\cos(\theta-\alpha)+e\cos(\theta)$ and passing through the point $(r_1,\theta_1)$. How should one go about solving this problem without making use of cartesian coordinates?
There seem to be too many constants in your equation for a line.

In Cartesian a line has a slope and an intercept.

In polar, the line has a direction, and a distance from the origin.

So I would make my line:
$r = l \csc (\theta + \alpha)$  Where $\alpha$ is the direction, and $l$ is the distance.

And, $\theta = \alpha$ for lines through the origin.

Or, $r = l \sec (\theta + \alpha)$ where alpha is the normal.

Now in your example which variables are we tuning?  There are infinitely many lines that go through ($r_1, \theta_1$) So, what is the direction of our line.  

Lets assume that the direction is fixed, using $\alpha$ is normal to our line.

What is the distance of a line perpendicular to $\alpha$ that goes through ($r_1, \theta_1$)?

$l = r_1 \cos (\alpha - \theta_1)$ 

$r = r_1 \cos (\alpha - \theta_1) \sec (\theta + \alpha)$
There seem to be too many constants in your equation for a line.

In Cartesian a line has a slope and an intercept.

In polar, the line has a direction, and a distance from the origin.

So I would make my line:
$r = l \csc (\theta + \alpha)$  Where $\alpha$ is the direction, and $l$ is the distance.

And, $\theta = \alpha$ for lines through the origin.

Or, $r = l \sec (\theta + \alpha)$ where alpha is the normal.

Now in your example which variables are we tuning?  There are infinitely many lines that go through ($r_1, \theta_1$) So, what is the direction of our line.  

Lets assume that the direction is fixed, using $\alpha$ is normal to our line.

What is the distance of a line perpendicular to $\alpha$ that goes through ($r_1, \theta_1$)?

$l = r_1 \cos (\alpha - \theta_1)$ 

$r = r_1 \cos (\alpha - \theta_1) \sec (\theta + \alpha)$