Question

Write a polar equation of a conic with the focus at the

origin and the given data \({\rm{Parabola, directrix x = - 3}}\).

Short answer

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

Step-by-step solution

Derivation.

\(\begin{aligned}{l}{\rm{When the attention is on the origin, the eccentricity is magnified e}} \ne {\rm{1,}}\\{\rm{as well as the directrix x = \pm d,}}\\{\rm{Then these are the two conics that meet these conditions:}}\\{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm ecos\theta }}}}\\{\rm{When the attention is on the origin, the eccentricity is magnified e}} \ne {\rm{1,}}\\{\rm{as well as the directrix}}\,{\rm{y = \pm d,}}\\{\rm{Then these are the two conics that meet these conditions:}}\\{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm esin\theta }}}}\\{\rm{Here}}\\{\rm{e = 1 There is only one conic for each given pair of focus and directrix, }}\\{\rm{and you must determine which sign to use in the denominator}}{\rm{.}}\end{aligned}\)

}\)Step 2

\({\rm{Due to the fact that the directrix is x = - 3 and d = 3 and also cos\theta in the lowest common denominator}}\)

\({\rm{Furthermore, because conic is a parabola, this implies e = 1}}{\rm{.}}\)

\(\begin{aligned}{l}{\rm{r = }}\frac{{{\rm{(1) \times 3}}}}{{{\rm{1 \pm (1)cos\theta }}}}\\{\rm{r = }}\frac{{\rm{3}}}{{{\rm{1 \pm cos\theta }}}}\end{aligned}\)

Evaluation.

\(\begin{aligned}{l}{\rm{Because there is only one parabola that corresponds to a pair of focus and directrix,}}\\{\rm{ the proper sign in the denominator must be determined}}{\rm{.}}\end{aligned}\)

\({\rm{Take note of:}}\)

\({\rm{(1) Because the directrix is a vertical line, the parabola is horizontal}}{\rm{.}}\)

\(\begin{aligned}{l}{\rm{(2) Because the directrix lies to the left of the origin, the parabola opens to the right,}}\\{\rm{ because the directrix and a parabola cannot intersect}}{\rm{.}}\end{aligned}\)

\(\begin{aligned}{l}{\rm{That is to say r(\pi )is limited and r(0) }}\\{\rm{This is only valid for the parabola shown below}}\\{\rm{ (among the two cases we have to decide between)}}\end{aligned}\)

\({\rm{r = }}\frac{{\rm{3}}}{{{\rm{1 - cos\theta }}}}\).