Question

# The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph: displaystyle{e}={2}, directrix displaystyle{r}=-{2} sec{theta}.

Conic sections
The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:
$$\displaystyle{e}={2},$$
$$directrix \displaystyle{r}=-{2} \sec{\theta}.$$

2021-03-10
$$\displaystyle{e}={2},{r}=-{2} \sec{\theta}$$
Since $$\displaystyle{e}>{1}$$, therefore the conic is a hyperbola.
Now consider the directrix,
$$\displaystyle{r}=-{2} \sec{\theta}$$
$$\displaystyle\Rightarrow{r}=-\frac{2}{{ \cos{\theta}}}$$
$$\displaystyle\Rightarrow{r} \cos{\theta}=-{2}$$
$$\displaystyle\Rightarrow{x}=-{2}$$
Now comparing whith $$\displaystyle{x}=-{p},$$ we get
$$\displaystyle{p}={2}$$
Therefore equation is,
$$\displaystyle{r}=\frac{{{e}{p}}}{{{1}-{e} \cos{\theta}}}$$
$$\displaystyle\Rightarrow{r}=\frac{{{2}\cdot{2}}}{{{1}-{2} \cos{\theta}}}$$
$$\displaystyle\Rightarrow{r}=\frac{4}{{{1}-{2} \cos{\theta}}}$$
Now the graph is,