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 step by step guidance

1. Identify the form of the polar equation given: \(r = \frac{1}{2 - 2 \sin \theta}\). This resembles the general form for conic sections in polar coordinates: \(r = \frac{ed}{1 + e \cos \theta}\) or \(r = \frac{ed}{1 + e \sin \theta}\), where \(e\) is the eccentricity and \(d\) is the distance from the pole to the directrix.
2. Rewrite the denominator to match the standard form. Notice that \(2 - 2 \sin \theta = 2(1 - \sin \theta)\), so the equation can be expressed as \(r = \frac{1}{2(1 - \sin \theta)} = \frac{\frac{1}{2}}{1 - \sin \theta}\). This matches the form \(r = \frac{ed}{1 - e \sin \theta}\), where \(e\) and \(d\) are positive constants.
3. From the rewritten form, identify the eccentricity \(e\) and the product \(ed\). Since the denominator is \(1 - e \sin \theta\), and comparing to \(1 - \sin \theta\), we see that \(e = 1\). Then, \(ed = \frac{1}{2}\), so \(d = \frac{1}{2}\).
4. 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.
5. 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 \(\sin \theta\) 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 \(\theta\) to the angle that minimizes \(r\).

Key Concepts

Polar Equations of Conic Sections

Polar equations describe conic sections using the radius r and angle θ from the pole. The general form r = ed / (1 ± e sin θ) or r = ed / (1 ± e cos θ) relates eccentricity e and directrix distance d, defining ellipses, parabolas, or hyperbolas based on e's value.

Eccentricity and Classification of Conics

Eccentricity (e) measures how much a conic deviates from being circular. If e < 1, the conic is an ellipse; if e = 1, a parabola; and if e > 1, a hyperbola. Identifying e from the equation helps classify the conic and understand its geometric properties.

Graphing and Identifying Key Features

Graphing polar conics involves plotting points for various θ values and labeling vertices, foci, and directrices. Vertices are points closest or farthest from the pole, foci are fixed points defining the conic, and directrices are lines related to eccentricity. Asymptotes appear only for hyperbolas.