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Chapter 6: Problem 24

Identify the conic and sketch its graph. \(r=\frac{5}{-1+2 \cos \theta}\)

Short Answer

Expert verified
The conic section represented by the polar equation \(r = \frac{5}{-1+2 \cos \theta}\) is a hyperbola with eccentricity e = 2 and semi-latus rectum p = -5. The graph of this hyperbola is centered at the origin in the polar coordinate plane.

Step by step solution

01

Determine the eccentricity e and semi-latus rectum p

The polar equation given is \(r = \frac{5}{-1+2 \cos \theta}\). Rewrite it as \(r = \frac{5}{1 - 2 \cos \theta}\) for the form \(r = \frac{ep}{1 卤 e \cos \theta}\). Comparing coefficients, the eccentricity e = 2 and the semi-latus rectum p = -5.
02

Identify the conic section with eccentricity e

Since the eccentricity e = 2 is greater than 1, the conic section is a hyperbola.
03

Plot the graph

Due to the complexity of hyperbolas, it's more manageable to plot the graph of this conic section using a graphing tool or software. Plot the function \(r = \frac{5}{1 - 2 \cos \theta}\) where r is the radius (distance from the origin) and \(\theta\) is the angle in polar coordinates. This will result in a hyperbola graph centered at the origin.

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