Given the following polar equation
![\[ r = \frac{1}{-\frac{1}{2}+ \cos \theta} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-129bd773b805df8fc165c9330dd532a6_l3.png "Rendered by QuickLaTeX.com")
for a conic section with a focus 
at the origin and vertical directrix lying to the right of , determine the eccentricity 
and the distance 
from the focus to the directrix.
By Theorem 13.18 we know the points on a conic section satisfy the polar equation
![\[ r = \frac{ed}{1 + e \cos \theta}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-dbff264e45f1e7d239bd0749f577f580_l3.png "Rendered by QuickLaTeX.com")
Therefore, in this case we have
![\[ r = \frac{1}{-\frac{1}{2} + \cos \theta} = \frac{2}{2 \cos \theta - 1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1d1b6b44199acff70add52a28851c0a1_l3.png "Rendered by QuickLaTeX.com")
Therefore, 
and 
.