# How do you name the curve given by the conic r=6/(2+sintheta)?

Aug 22, 2016

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 $\theta = 0$, is

$\frac{l}{r} = 1 + e \cos \theta$, where

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

(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

$\frac{3}{r} = 1 + \frac{1}{2} \cos \theta$, So, e = 1/2 < 1, and so, the conic is an

ellipse. The semi major axis a is given by
 l = a(1-e^2)=a(1-1/4) =3a/4=3. So, a = 1/4#