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

Answer 1

Benjamin Achtenberg

Hyperbola.

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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Answer 2

Sadie Ackerman

The curve given by the conic ( r = \frac{4}{1 + 2 \sin(\theta)} ) is known as a cardioid.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.