Question

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π. r = 3/(1 - cos θ)

Accepted Answer

Recognize that the given equation $$r = \frac{3}{1 - \cos 	heta} is a $$polar equation representing a conic section. Since the denominator is of the form $$1 - e \cos 	heta$$ with $$e = 1$$, this corresponds to a parabola in polar coordinates with the focus at the pole. Identify the key features of the curve by analyzing the denominator $$1 - \cos 	heta. $$Note that when $$\cos 	heta$$ approaches 1, the denominator approaches zero, causing $$r to $$become very large, indicating a vertical asymptote or a direction where the curve extends to infinity. Create a table of values for $$	heta at $$important points such as $$0$$, $$\frac{\pi}{2}$$, $$\pi$$, $$\frac{3\pi}{2}$$, and $$2\pi. $$Calculate the corresponding $$r$$ values (without final numeric evaluation) to understand the shape and direction of the curve at these angles. Plot the points $$(r, 	heta) on $$the polar coordinate plane using the values from the table. Mark these points clearly and draw the curve smoothly through them, keeping in mind the behavior near the asymptote where $$r$$ becomes very large. Add arrows along the curve to indicate the direction of increasing $$	heta$$ from $$0 to$$ $$2\pi. $$Label key points corresponding to the angles used in the table to show how the curve is generated as $$	heta$$ increases.

63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π.

r = 3/(1 - cos θ)

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Key Concepts

Polar Coordinates and Graphing

Conic Sections in Polar Form

Parametric Tracing and Direction of Curves