Question

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

Answer 1

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, \(\displaystyle{x}=-{4}\)