# Identify the conic section given by the polar equation displaystyle{r}=frac{4}{{{1}- cos{theta}}} and also determine its directrix.

Question
Conic sections
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.

2021-02-13
Step 1
We convert the equation into rectangular form first
$$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$
$$\displaystyle{r}{\left({1}- \cos{\theta}\right)}={4}$$
$$\displaystyle{r}-{r} \cos{\theta}={4}$$
$$\displaystyle\sqrt{{{x}^{2}+{y}^{2}}}-{x}={4}$$
$$\displaystyle\sqrt{{{x}^{2}+{y}^{2}}}={x}+{4}$$
$$\displaystyle{x}^{2}+{y}^{2}={\left({x}+{4}\right)}^{2}$$
$$\displaystyle{x}^{2}+{y}^{2}={x}^{2}+{8}{x}+{16}$$
$$\displaystyle{y}^{2}={8}{x}+{16}$$
$$\displaystyle{y}^{2}={8}{\left({x}+{2}\right)}$$
This is a parabola.
Step 2
Compare the equation with the standard form
$$\displaystyle{\left({y}-{k}\right)}^{2}={4}{p}{\left({x}-{h}\right)}$$
$$\displaystyle{h}=-{2},{k}={0}$$
$$\displaystyle{8}{p}={2}$$
$$\displaystyle{\quad\text{or}\quad},{p}={2}$$
Directrix $$\displaystyle={x}={h}-{p}{\quad\text{or}\quad}{x}=-{2}-{2}{\quad\text{or}\quad}{x}=-{4}$$
Answer: Parabola, $$\displaystyle{x}=-{4}$$

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