Dillan Gibbs

2022-02-12

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

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$\mathrm{cos}\theta =\mathrm{sin}(\frac{\pi}{2}-\theta )$ ,

transforming$(\frac{\pi}{2}-\theta )\to \theta$ .

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

$l=4=a({e}^{2}-1)=3a$ , and so,

the semi major axis the hyperbola$a=\frac{4}{3}$ .

As

transforming

we get the equation in the given form

This transformation is rotation of the initial line through

For

Here

the semi major axis the hyperbola

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