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Chapter 10: Problem 52

Write a polar equation of the conic that is named and described. Hyperbola: a focus at the pole; directrix: \(x=-1 ; e=\frac{3}{2}\)

Short Answer

Expert verified
The polar equation of the given hyperbola is \(r = \frac{\frac{3}{2}*1}{1+\frac{3}{2}\cos(\theta)}\).

Step by step solution

01

Identify the Given Properties

The given properties of the hyperbola are: focus at the pole (origin), directrix at x=-1, and eccentricity e=3/2.
02

Determine Perpendicular Distance from the Pole to the Directrix

The pole is at the origin (0,0), and the directrix is a vertical line at x=-1. The perpendicular distance \(d\) between a point and a line in a coordinate geometry is simply the absolute difference along the axis. So \(d = |-1-0| = 1\).
03

Formulate Polar Equation of a Hyperbola

Now plug the given eccentricity \(e=3/2\) and computed distance \(d=1\) into the standard form of the polar equation of a conic section \(r = \frac{ed}{1+e\cos(\theta)}\). The polar equation becomes \(r = \frac{\frac{3}{2}*1}{1+\frac{3}{2}\cos(\theta)}\).

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Most popular questions from this chapter

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{16}{3+5 \cos \theta} $$

Describe a strategy for graphing \(r=\frac{1}{1+\sin \theta}\)

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(j^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{8}{2+2 \sin \theta} $$

Use a graphing utility to graph the equation. Then answer the given question. $$ \begin{aligned} &r=\frac{4}{1-\sin \left(\theta-\frac{\pi}{4}\right)} ; \text { How does the graph differ from the }\\\ &\text { graph of } r=\frac{4}{1-\sin \theta} ? \end{aligned} $$
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