Polar equation of conic?

Question

It says write a polar equation of a conic with the focus at the origin and the given data.

It is an ellipse, eccentricity = 1/2, and directrix r= 4secθ

I don't get how to use the given directrix, can anyone help?

Answer 1

It says write a polar equation of a conic with the focus at the origin and the given data.

It is an ellipse, eccentricity = 1/2, and directrix r= 4secθ.
__________

Let
d = distance from center of ellipse to directrix.

e = c/a = a/d = 1/2

a = 2c
d = 2a

d = 4c

The directrix is r = 4secθ or x = 4.

The distance between the focus and directrix is:

d - c = 4 - 0 = 4
4c - c = 4
3c = 4
c = 4/3
a = 2c = 8/3

The equation of the ellipse is:

r = a(1 - e²)/(1 - ecosθ)
r = (8/3)[1 - (1/2)²] / [1 + (1/2)cosθ]
r = (8/3)[1 - 1/4] / [1 + (1/2)cosθ]
r = (8/3)(3/4) / [1 + (1/2)cosθ]

r = 2 / [1 + (1/2)cosθ]

Answer 2

The directrix is x = a/e in cartesian.

r = 4secθ is the same as rcosθ = 4.

but x = rcosθ so the directrix is x = 4

a/e = 4 and with e = ½ means a = 2

b² = a²(1 - e²) = 4(1 - 1/4) = 3

The equation of your ellipse is x² / 4 + y² / 3 = 1 and changing to polar is

r² cos² θ /4 + r² sin²θ / 3 = 1

r² (3cos²θ + 4sin²θ) = 12

r² = 12/( 3 + sin² θ)