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Chapter 6: Problem 22

Identify the conic and sketch its graph. \(r=\frac{9}{3-2 \cos \theta}\)

Short Answer

Expert verified
The given equation represents an ellipse with an eccentricity of 0.5.

Step by step solution

01

Identify the conic section

The equation given is \(r=\frac{9}{3-2 \cos \theta}\). Recognize that this is the polar form of a conic section \(\frac{p}{1+e \cos \theta}\) or \(\frac{p}{1-e \cos \theta}\). Comparing the given equation to the templates, we realize it fits the template of an ellipse, provided that \(e < 1\) for an ellipse.
02

Calculate eccentricity

Let's calculate the eccentricity, \(e\). The denominator is \((3-2 \cos \theta)\), and if we compare it to \((1-e \cos \theta)\), we find \(e = \frac{1}{2}\) which confirms our assumption that it's an ellipse since \(e <1\).
03

Graph the conic section

To graph an ellipse in polar coordinates, you draw the major axis along the horizontal axis (from \(-e \cdot p\) to \(p\)), the minor axis is perpendicular to the major axis at the center of the ellipse. Ellipses in polar form always have one focus at the origin. Note that due to the eccentricity value, the ellipse will be wider (major axis > minor axis). You sketch it using the relationship between \(r\), \(\theta\) and eccentricity.

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Most popular questions from this chapter

Find an equation of the tangent line to the parabola at the given point, and find the \(x\) -intercept of the line. \(y=-2 x^{2},(-1,-2)\)

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. \(3 x^{2}+y^{2}+18 x-2 y-8=0\)

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