Question

Polar CoordinatesExercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.e = 1/3, r sin θ = −6

Accepted Answer

Recall the general polar form of a conic section with one focus at the origin:

$$ r = \frac{ed}{1 + e \cos 	heta} \quad 	ext{or} \quad r = \frac{ed}{1 + e \sin 	heta} $$

where $$e is $$the eccentricity and $$d is $$the distance from the focus to the directrix. Identify the given eccentricity $$e = \frac{1}{3}$$ and the directrix equation $$r \sin 	heta = -6. $$Since the directrix involves $$r \sin 	heta$$, it corresponds to a horizontal line in polar coordinates, so the conic equation will use the $$\sin 	heta$$ form. Rewrite the directrix equation $$r \sin 	heta = -6 to $$find the distance $$d$$ from the origin to the directrix. Note that $$r \sin 	heta$$ represents the $$y-$$coordinate in Cartesian coordinates, so the directrix is the line $$y = -6. $$Thus, $$d = 6 ($$distance is positive). Substitute $$e = \frac{1}{3}$$ and $$d = 6$$ into the polar conic formula with $$\sin 	heta$$:

$$ r = \frac{e d}{1 + e \sin 	heta} = \frac{\frac{1}{3} 	imes 6}{1 + \frac{1}{3} \sin 	heta} $$ Simplify the expression to write the final polar equation of the conic section in terms of $$r$$ and $$	heta.$$

Polar Coordinates

Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.

e = 1/3, r sin θ = −6

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Key Concepts

Polar Coordinates and Polar Equations

Eccentricity of Conic Sections

Directrix in Polar Form