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.

Caelan 2020-11-05 Answered
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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Expert Answer

Laaibah Pitt
Answered 2020-11-06 Author has 15861 answers
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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Answered 2021-10-26 Author has 1944 answers

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