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

polar-equation

#1

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:

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

dd = y - 1 ; df = sqrt (x^2+y^2)

We want df / dd = 1/4 so:

sqrt (x^2+y^2)/ (y - 1) = 1/4

4√(x2 + y2) = y - 1

4r = r sin θ - 1

(sin θ - 4) r = 1

r = 1/(sin θ - 4)