determine the type of conic

Question

r=2/1-2sin(theta)

any help would be wonderful thx

Answer 1

The polar equation for a conic: for a conic with a focus at the origin, if the directrix is y=± p, where p is a positive real number , and the eccentricity is a positive real number e , the conic has a polar equation

r = e p / ( 1 ± e sin θ )

r = 2 / 1 - 2 sin θ

Compare

e = 2

e p = 2

therefore p = 1

Because sine is in the denominator the directrix is y = - p = -1

and because e > 1 , the conic is hyperbola.