Finding Directrix and Sketching Hyperbola in Polar Coordinates

[r34racer01]

Homework Statement

Hyperbola formula 9x^2 - 4y^2 + 36x + 24y - 36 = 0.
Convert to rectangular form, find coordinates of the vertices, find coordinates of the foci, find eccentricity, what is the equation of the conic section in polar coordinates if the pole is taken to be the leftmost focus? Sketch.

Homework Equations

a^2 +b^2 = c^2 directrix: x = (a^2)/c e = c/a

The Attempt at a Solution

So I have the hyperbola formula 9x^2 - 4y^2 + 36x + 24y - 36 = 0 I converted it to...
((x+2)^2)/4 - ((y-3)^2)/9 = 1
The vertices I have as (0,3) and (-4,3)
The foci I have as (-2+ sq. root 13,3) and (-2 + sq. root 13, 3)
Eccentricity is (sq. root 13)/2
Now here's the problem I'm having issues trying to find the directrix. I found a formula that says its x = (a^2)/c so after plugging in I get 1.109 but that's between the right vertex and right focus, aren't the directrixes suppose to be between the center and the vertices. Also I'm not quite sure what to do "the pole is taken to be the leftmost focus."

 

[kurt.physics]

The equation of the conic section in polar coordinates is r = (sq. root 13)/2 *cos(theta)I'm having a hard time sketching this out since I can't find the directrixes. If anyone could help with that and explain how to do the pole taken to be the leftmost focus that would be awesome!