Question

Write a polar equation of a conic with the focus at the origin and the given data.

\({\rm{ Hyperbola, eccentricity 3, directrix x = 3}}\)

Short answer

Polar equation of a conic with the focus at the origin and the Hyperbola, eccentricity\({\rm{e = 3}}\), directrix\({\rm{x = 3}}\) and \({\rm{r = }}\frac{{\rm{9}}}{{{\rm{1 + 3cos\theta }}}}\).

Step-by-step solution

Hyperbola and directrix.

Hyperbola with eccentricity\({\rm{e = 3}}\)

Directrix\({\rm{x = 3}}\)

\({\rm{x = 3}}\)

Hyperbola and ellipse.

Half of the hyperbola and or ellipse that was closest to the focus at the origin.

If the directrix is at\({\rm{x = 3 }}\)then the part of the hyperbola closest to this focus opens to the left and the form of the equation to use

\(\begin{aligned}{l}{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 + ecos\theta }}}}\\{\rm{r = }}\frac{{{\rm{(3)(3)}}}}{{{\rm{1 + (3)cos\theta }}}}\\{\rm{r = }}\frac{{\rm{9}}}{{{\rm{1 + 3cos\theta }}}}\end{aligned}\)

Polar equation of a conic with the focus at the origin is.

\({\rm{r = }}\frac{{\rm{9}}}{{{\rm{1 + 3cos\theta }}}}\)