Polar Equations??

Question

1)r=(12)/(3+4 cos theta)
2) r=(23)/(7-5sin theta)

a) its directrix is vertical or hortizontal? why?
b) its directrix is above the pole, below the poler, to the right of the pole, to the left of the pole? why?

Answer 1

1)
let Θ take the values 0°, 90°, 180°, and 270°, and plot points. you get (12/7, 0), (4, 90), (-12, 180), and (4, 270). that should be enough to tell you you've got a hyperbola, left focus at the pole, directrix vertical in between the 12/7 and the 12 on the 0° axis. more specifically, if you rewrite the equation as 4/(1 + 4/3 cos Θ), you know 4/3 is the eccentricity, and since it's greater than 1, you've got a hyperbola. in this form, the 4 on top is the semi-latus rectum. factor the 4 into (4/3)(3), and the 3 is the distance from the pole to the directrix.

2) plotting, you have (23/7, 0), (23/2, 90), (23/7, 180), (23/12, 270), an ellipse. directrix horizontal below the pole. eccentricity 5/7, pole to directrix distance = 23/5.