How do you name the curve given by the conic r=41+2sin⁡θ?

Solved: "How do you name the curve given by..."

Dillan Gibbs

2022-02-12

How do you name the curve given by the conic

r = \frac{4}{1 + 2 \sin θ}

dicky23628u6a

Beginner2022-02-13Added 12 answers

The polar equation

r = \frac{l}{1 + e \cos θ}

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

θ = 0

.
As

\cos θ = \sin (\frac{π}{2} - θ)

,
transforming

(\frac{π}{2} - θ) \to θ

.
we get the equation in the given form
This transformation is rotation of the initial line through

\frac{π}{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) = 3 a

, and so,
the semi major axis the hyperbola

a = \frac{4}{3}

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