Question

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 1/(2 - 2 sin θ)

Accepted Answer

Identify the form of the polar equation given: $$r = \frac{1}{2 - 2 \sin 	heta}. $$This resembles the general form for conic sections in polar coordinates: $$r = \frac{ed}{1 + e \cos 	heta} or$$ $$r = \frac{ed}{1 + e \sin 	heta}$$, where $$e is $$the eccentricity and $$d is $$the distance from the pole to the directrix. Rewrite the denominator to match the standard form. Notice that $$2 - 2 \sin 	heta = 2(1 - \sin 	heta)$$, so the equation can be expressed as $$r = \frac{1}{2(1 - \sin 	heta)} = \frac{\frac{1}{2}}{1 - \sin 	heta}. $$This matches the form $$r = \frac{ed}{1 - e \sin 	heta}$$, where $$e$$ and $$d$$ are positive constants. From the rewritten form, identify the eccentricity $$e$$ and the product $$ed. $$Since the denominator is $$1 - e \sin 	heta$$, and comparing to $$1 - \sin 	heta$$, we see that $$e = 1. $$Then, $$ed = \frac{1}{2}$$, so $$d = \frac{1}{2}. $$Determine the type of conic based on the eccentricity $$e. $$Since $$e = 1$$, the conic is a parabola. This means the curve has a single focus at the pole and a directrix line located at a distance $$d$$ from the pole. Label the key features on the graph: the focus is at the pole (origin), the directrix is the horizontal line $$y = -d ($$since the equation involves $$\sin 	heta$$ and the sign is negative), and the vertex is the point on the curve closest to the pole. Use the equation to find the vertex by setting $$	heta to $$the angle that minimizes $$r.$$

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 1/(2 - 2 sin θ)

Verified video answer for a similar problem:

Key Concepts

Polar Equations of Conic Sections

Eccentricity and Classification of Conics

Graphing and Identifying Key Features

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.r = 1/(2 - 2 sin θ)

Verified video answer for a similar problem:

Key Concepts

Polar Equations of Conic Sections

Eccentricity and Classification of Conics

Graphing and Identifying Key Features

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 1/(2 - 2 sin θ)