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Chapter 8: Problem 30

Convert the polar equation to rectangular form and sketch its graph. $$ r=5 \cos \theta $$

Short Answer

Expert verified
The polar equation \( r = 5\cos\theta \) converts to the rectangular form \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \), which represents a circle with radius 2.5 centered at \( ({5 \over 2}, 0) \).

Step by step solution

01

Converting the polar equation to rectangular form

We start with the polar equation \( r = 5\cos\theta \). Multiplying both sides of the equation by r yields \( r^2 = 5r\cos\theta \). We then substitute \( r = \sqrt{x^2 + y^2} \) and \( \cos\theta = {x \over r} \) into the equation to get \( x^2 + y^2 = 5x \). Rearranging terms, we have \( x^2 - 5x + y^2 = 0 \). Completing the square on x results in the rectangular equation \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \).
02

Sketching the graph

The rectangular equation \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \) describes a circle with radius \( {5 \over 2} \) and center at \( ({5 \over 2}, 0) \). So, the sketch should look like a circle at point 2.5 on the x-axis and with radius 2.5

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Most popular questions from this chapter

Use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=\theta, \quad 0 \leq \theta \leq \pi $$

Identify each conic. (a) \(r=\frac{5}{1-2 \cos \theta}\) (b) \(r=\frac{5}{10-\sin \theta}\) (c) \(r=\frac{5}{3-3 \cos \theta}\) (d) \(r=\frac{5}{1-3 \sin (\theta-\pi / 4)}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \sin \theta}\)

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sin 2 \theta, y=2 \cos 2 \theta $$
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