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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Recall the general polar form of a conic section with one focus at the origin: \[ r = \frac{ed}{1 + e \cos \theta} \quad \text{or} \quad r = \frac{ed}{1 + e \sin \theta} \] 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 \(r \sin \theta = -6\). Since the directrix involves \(r \sin \theta\), it corresponds to a horizontal line in polar coordinates, so the conic equation will use the \(\sin \theta\) form.

Rewrite the directrix equation \(r \sin \theta = -6\) to find the distance \(d\) from the origin to the directrix. Note that \(r \sin \theta\) 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 \(\sin \theta\): \[ r = \frac{e d}{1 + e \sin \theta} = \frac{\frac{1}{3} \times 6}{1 + \frac{1}{3} \sin \theta} \]

Simplify the expression to write the final polar equation of the conic section in terms of \(r\) and \(\theta\).

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Polar Coordinates and Polar Equations

Polar coordinates represent points using a radius and an angle (r, θ) from the origin. Polar equations express curves in terms of r and θ, which is essential for describing conic sections centered at a focus located at the origin.

Eccentricity of Conic Sections

Eccentricity (e) measures how much a conic section deviates from being circular. For conics with a focus at the origin, e < 1 indicates an ellipse, e = 1 a parabola, and e > 1 a hyperbola. This value helps determine the shape and equation of the conic.

Directrix in Polar Form

The directrix is a fixed line used to define conics relative to a focus. In polar coordinates, the directrix can be expressed as a linear equation involving r and θ, such as r sin θ = -6, which is used along with eccentricity to find the polar equation of the conic.

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