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Chapter 6: Problem 26

Identify the conic and sketch its graph. \(r=\frac{3}{2+6 \sin \theta}\)

Short Answer

Expert verified
The given polar equation \(r=\frac{3}{2+6 \sin \theta}\) represents a hyperbola.

Step by step solution

01

Identifying the Conic

The given equation is \(r = \frac{3}{2 + 6 \sin \theta}\). This equation looks similar to the standard form of a conic in polar coordinates: \(r = \frac{p}{1 \pm e \sin \theta}\) or \(r = \frac{p}{1 \pm e \cos \theta}\), where \(r\) is the distance from the origin to a point on the conic, \(p\) is the distance from the focus to the directrix, \(e\) is the eccentricity and \(\theta\) is the angle formed with the positive x-axis. Comparing the given equation to this standard form, it can be seen that \(p = 3\), and \(e = 6\), and because there's a addition in the denominator, it must represent a conic section where \(e < 1\) or \(e > 1\). Because \(e > 1\), it represents a hyperbola.
02

Graphing the Conic

To sketch the graph, it's easier to plot some representative points. Lets consider \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) and \( 2\pi\). Calculate their corresponding \(r\) values using the given equation. Afterwards, plot these points in a graph and draw a smooth curve that passes all the points to represent the graph of the hyperbola.

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