How do you name the curve given by the conic r=62+sin⁡θ?

Question

Bleadafat9om · Accepted Answer

Ellipse
Explanation:
The polar equation of a conic, referred to a/the focus as the pole

r = 0

and the perpendicular from the pole to the (corresponding)
directrix as the initial line

θ = 0

, is

\frac{l}{r} = 1 + e \cos θ

, where
e is the eccentricity of the conic and l is the semi latus rectum = #

(\frac{1}{2})

X (length of the chord of the conic through the focus,that is perpendicular to the initial line).
The conic is named an ellipse, parabola or hyperbola according as #

e < = > 1 .

#
Interestingly, for the circle the focus is at the center and

l =

radius a and

e = 0

. The equation is simply

r = a

.
Here, the equation is

\frac{3}{r} = 1 + \frac{1}{2} \cos θ

. So,

e = \frac{1}{2} < 1

, and so, the conic is an ellipse. The semi major axis a is given by

l = a (1 - e^{2}) = a (1 - \frac{1}{4}) = 3 \frac{a}{4} = 3

. So,

a = \frac{1}{4}

.

How do you name the curve given by the conic

Answered question

Answer & Explanation

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