Belen Jones

2022-02-10

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

Bleadafat9om

Beginner2022-02-11Added 13 answers

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

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

Explanation:

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

directrix as the initial line

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

The conic is named an ellipse, parabola or hyperbola according as #

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

Here, the equation is