How do I find the pole and directrix? And then what do I do to find the distance between them? - Thanks.
To determine the distance between the pole and the directrix, you do not need to find the pole and directrix.
The polar form of equation of a conic (for horizontal directrix) can be written as:
r = ` ke/(1+-e sintheta)`
where, k is the distance between the pole and the directrix
We can rewrite the given equation of r to this format.
r = `4/(-2-6 sintheta)`
Dividing the numerator and denominator by -2, we get
r = `-2/(1+3 sintheta)`
Comparing with the standard equation, e = 3 and k = 2/3 .
Thus, the distance between the pole and the directrix for the given case is 2/3.