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Chapter 6: Problem 70

Determine whether the statement is true or false. Justify your answer. The conic represented by the following equation is a parabola. \(r=\frac{6}{3-2 \cos \theta}\)

Short Answer

Expert verified
The given statement is false. The conic represented by the equation \(r=\frac{6}{3-2 \cos \theta}\) is a hyperbola, not a parabola.

Step by step solution

01

Identify the general form of polar equations

The general form of the polar equations of conics is \(r=\frac{p}{1+e cos(\theta -\phi)}\), where \(e\) is the eccentricity of the conic.
02

Compare the given equation with the general form

Looking at the given equation \(r=\frac{6}{3- 2 \cos \theta}\), rewrite it as \(r=\frac{6}{1+2(1 - \cos \theta)}\). By comparing it with the general form, it can be observed that \(p=6\) and \(e=2\).
03

Determine the type of conic

The eccentricity \(e\) determines the type of conic. If \(e=0\), the conic is a circle. If \(01\), the conic is a hyperbola. Here, \(e=2\) which is greater than 1. So, the conic is a hyperbola, not a parabola.

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Most popular questions from this chapter

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(x^{2}+5 y^{2}-8 x-30 y-39=0\)

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Find the standard form of the equation of the parabola with the given characteristics. Vertex: (0,2)\(;\) directrix: \(y=4\)
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