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Q 36.

Expert-verified

Calculus

Found in: Page 731

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.
$r = 2 \cos θ$

The required equation is ${(x - 1)}^{2} + y^{2} = 1$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given equation in polar coordinates is:

$r = 2 \cos θ$

Step 2. Find the equation in rectangular coordinates.

$r = 2 \cos θ$

First, multiply both sides by r,

$r^{2} = 2 r \cos θ r^{2} = 2 x [Since r \cos θ = x] x^{2} + y^{2} = 2 x [S i n c e r^{2} = x^{2} + y^{2}]$

Now add $- 2 x$ on both sides of the equation,

role="math" localid="1649264450788" $x^{2} + y^{2} - 2 x = 2 x - 2 x x^{2} + y^{2} - 2 x = 0$

Complete the square in x,

${(x - 1)}^{2} + y^{2} - 1 = 0$

now add 1 on both sides of the equation,

${(x - 1)}^{2} + y^{2} - 1 + 1 = 0 + 1 {(x - 1)}^{2} + y^{2} = 1$

Therefore, the equation in rectangular coordinates is ${(x - 1)}^{2} + y^{2} = 1$ .

Most popular questions for Math Textbooks

Consider the hyperbola with equation $\frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1 .$ Let F be the focus with coordinates $(\sqrt{A^{2} + B^{2}}, 0) .$ Let $e = \frac{\sqrt{A^{2} + B^{2}}}{A}$ and l be the vertical line with equation $x = \frac{A}{e} .$ Show that for any point P on the hyperbola, $\frac{F P}{D P} = e,$ where D is the point on l closest to P.

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.

$r = \frac{α}{1 + \sin θ}$

In Exercises 48–55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$x = 0$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

One petal of the polar rose $r = \cos 4 θ$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

one petal of the polar rose $r = \cos 3 θ$

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