Belen Jones

2022-02-10

How do you name the curve given by the conic $r=\frac{6}{2+\mathrm{sin}\theta }$?

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
$\frac{l}{r}=1+e\mathrm{cos}\theta$, where
e is the eccentricity of the conic and l is the semi latus rectum = #
$\left(\frac{1}{2}\right)$ 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
$\frac{3}{r}=1+\frac{1}{2}\mathrm{cos}\theta$. So, $e=\frac{1}{2}<1$, and so, the conic is an ellipse. The semi major axis a is given by
$l=a\left(1-{e}^{2}\right)=a\left(1-\frac{1}{4}\right)=3\frac{a}{4}=3$. So, $a=\frac{1}{4}$.

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