# How do you name the curve given by the conic r=4/(1+2sintheta)?

Aug 13, 2016

Hyperbola.

#### Explanation:

The polar equation $r = \frac{l}{1 + e \cos \theta}$ 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 $\cos \theta = \sin \left(\frac{\pi}{2} - \theta\right)$,

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 $e = 2 \mathmr{and}$

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

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

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