Find a polar equation of the conic

Question

Find a polar equation of the conic in terms of r with its focus at the pole.

 

Conic: Ellipse

Eccentricity: e=1/4

Directrix: y=1

 

r=?

Answer 1

Let (x,y) be on the ellipse. The distances from the directrix and focus respectively are:

 

  d_d = y - 1 ; d_f = √(x² + y²)

 

We want d_f / d_d = 1/4 so:

 

√(x² + y²) / (y - 1) = 1/4

 

4√(x² + y²) = y - 1

 

4r = r sin θ - 1

 

(sin θ - 4) r = 1

 

r = 1/(sin θ - 4)