Question

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

origin and the given data \({\rm{Ellipse, eccentricity }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{, directrix x = 4}}\).

Short answer

\({\rm{r = }}\frac{{\rm{4}}}{{{\rm{2 \pm cos\theta }}}}\)

Step-by-step solution

Derivation.

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

Evaluation.

\(\begin{aligned}{l}{\rm{Because the directrix is x = 4}},{\rm{d = 4 }}and\cos \theta ,\\deno\min ators.\\{\rm{It's also worth noting that e = 1 / 2}}\\r = \frac{{(1/2) \cdot 4}}{{1 \pm (1/2)\cos \theta }}\\{\rm{Both the numerator and the denominator should be multiplied by 2}}{\rm{.}}\\r = \frac{4}{{2 \pm \cos \theta }}\end{aligned}\)

Therefore, the result is\({\rm{r = }}\frac{{\rm{4}}}{{{\rm{2 \pm cos\theta }}}}\).