Dillan Gibbs

## Answered question

2022-02-12

How do you name the curve given by the conic $r=\frac{4}{1+2\mathrm{sin}\theta }$?

### Answer & Explanation

dicky23628u6a

Beginner2022-02-13Added 12 answers

The polar equation $r=\frac{l}{1+e\mathrm{cos}\theta }$ represents a conic whose eccentricity is e. A focus is the pole and the line from the pole away from the center is the initial line $\theta =0$.
As $\mathrm{cos}\theta =\mathrm{sin}\left(\frac{\pi }{2}-\theta \right)$,
transforming $\left(\frac{\pi }{2}-\theta \right)\to \theta$.
we get the equation in the given form
This transformation is rotation of the initial line through
$\frac{\pi }{2}$, about the pole, in the clockwise sense.
For $e>1$, the conic is named a hyperbola..
Here $e=2$ and
$l=4=a\left({e}^{2}-1\right)=3a$, and so,
the semi major axis the hyperbola $a=\frac{4}{3}$.

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