Question

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

Answer 1

Hyperbola.

Explanation:

The polar equation `r = 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 (pi/2-theta)`,

transforming `(pi/2-theta) to theta`,

we get the equation in the given form

This transformation is rotation of the initial line through

`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 = 4/3.` .

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