Look at the denominator of the conic equation to find the directrix. It should be in one of four forms: 1 + e*cos(theta); 1 - e*cos(theta); 1 + e*sin(theta); or 1 - e*sin(theta).
The denominator of the conic equation tells you in which direction respective to the conic section's center the directrix of the conic section lies as well as whether it is vertical or horizontal. For the forms with a "cos" function, the directrix is vertical, and for forms with the "sin" function, it is horizontal. In addition, whether the second part of the denominator is being added to or subtracted from one tells you where the directrix is: if the second part of the denominator is being added, the directrix is moved in the positive direction --- up or right, depending on whether the directrix is vertical or horizontal; otherwise, the directrix is moved in the negative direction --- down or left depending on whether the directrix is vertical or horizontal.
For example, in the equation "r = 4/(1 -- 2cos(theta))," the directrix is vertical and to the left of the conic section.
Observe the function to find its eccentricity. Recall that the polar equation for a conic section is in the form of "r = e*d/denom," where "denom" varies as stated in the previous step. Find the value of "e" for the equation to obtain the eccentricity. For example, in the equation "r = 4/(1 -- 2cos(theta))," "e" is 2. Thus, the eccentricity of the conic section is 2.
Determine the type of conic section the equation is describing. This requires you to merely observe the value you found for "e." If "e" is zero, the conic section is a circle. If "e" is between zero and one, the conic section is an ellipse. If "e" is exactly one, the conic section is a parabola. And if "e" is greater than one, the conic section is a hyperbola. For example, in the equation "r = 4/(1 -- 2cos(theta))," you can observe that "e" is 2, implying that the conic section corresponding to this equation is a hyperbola.