Given:
Directrix x = -5 and eccentricity = 2
To find:
The polar equation of the conic.
Solution:
Eccentricity = 2 > 0
Therefore the conic must be a hyperbola.
Directrix is vertical (at x = -5) and the vertical directrix is located to the left of the pole.
So that the equation is of the form:
[tex]$r=\frac{e p}{1-e \cos \theta}[/tex]
Since the eccentricity of this hyperbola is 1
The distance between the pole and directrix is
p = |-5|= 5
Substitute these in the above equation.
[tex]$r=\frac{(2)(5)}{1-2 \cos \theta}[/tex]
[tex]$r=\frac{10}{1-2 \cos \theta}[/tex]
The polar equation of the conic is [tex]r=\frac{10}{1-2 \cos \theta}[/tex] hyperbola.
