# Solve, a.Determine the conic section of the polar equation displaystyle{r}=frac{8}{{{2}+{2} sin{theta}}} represents. b. Describe the location of a directrix from the focus located at the pole.

Solve, a.Determine the conic section of the polar equation $$\displaystyle{r}=\frac{8}{{{2}+{2} \sin{\theta}}}$$ represents. b. Describe the location of a directrix from the focus located at the pole.

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Laaibah Pitt
a)
Given: $$\displaystyle{r}=\frac{8}{{{2}+{2} \sin{\theta}}}$$
On simplification, we get
$$\displaystyle\Rightarrow{r}=\frac{8}{{{2}{\left({1}+ \sin{\theta}\right)}}}$$
$$\displaystyle\Rightarrow{r}=\frac{4}{{{1}+ \sin{\theta}}}$$
Now, comparing with $$\displaystyle{r}=\frac{{{e}{p}}}{{{1}+{e} \sin{\theta}}}$$, we get
$$\displaystyle{e}={1}$$
$$\displaystyle{e}{p}={4}\Rightarrow{1}\times{p}={4}\Rightarrow{p}={4}$$
Since $$\displaystyle{e}={1}$$, therefore the given equation is a parabola.
Step 2
b)
Equation of directrix is,
$$\displaystyle{y}={p}$$
$$\displaystyle\Rightarrow{y}={4}$$
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