Precalculus

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Question

Chapter 4, Problem 10CR

To determine

The center and the radius of the circle $x2+4x+y2−2y−4=0$ and graph the circle.

Expert Solution & Answer

Answer to Problem 10CR

Solution:

The center is $(−2,1)$ and the radius is $r=3$ . The graph of the circle is

Precalculus, Chapter 4, Problem 10CR , additional homework tip 1

Explanation of Solution

Given information:

The equation of the circle $x2+4x+y2−2y−4=0$

Explanation:

Here, the equation of the circle is $x2+4x+y2−2y−4=0$

$⇒(x2+4x)+(y2−2y)−4=0$ .

Now, complete the square of each expression in the parentheses

The quadratic equation $ax2+bx+c=0$ can make the complete square by adding $(b/2a)2$ on both sides of the equation

For $(x2+4x)$ , the value $(b/2a)2=(4/2)2=4$

For $(y2−2y)$ , the value $(b/2a)2=(−2/2)2=1$

So, add $4+1$ on both sides of the equation of the circle

$⇒(x2+4x+4)+(y2−2y+1)−4=4+1$ .

$⇒(x+2)2+(y−1)2−4=5$

$⇒(x+2)2+(y−1)2=5+4=9$

$⇒(x+2)2+(y−1)2=32$ .

The standard form of the equation of the circle with center $(h,k)$ and radius $r$ is $(x−h)2+(y−k)2=r2$

Compare the given equation with $(x−h)2+(y−k)2=r2$

$⇒h=−2, k=1, r=3$

Hence, the center is $(−2,1)$ and the radius is $r=3$

To draw the graph of the circle,

First, plot the center $(−2,1)$

Here, the radius is $r=3$ , locate four points on the circle by plotting points $3$ units to the left, to the right, up, and down from the center.

Now, draw the circle passing through these points.

The graph of the circle is given as

Precalculus, Chapter 4, Problem 10CR , additional homework tip 2

Chapter 4, Problem 9CR

Chapter 4, Problem 11CR

Chapter 4 Solutions

Precalculus

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Knowledge Booster

The center and the radius of the circle x 2 + 4 x + y 2 − 2 y − 4 = 0 and graph the circle.

Solution Summary: The author explains how to determine the center and radius of the circle and graph it. The quadratic equation ax2+bx+c=0 can make the complete square.

The center and the radius of the circle $x2+4x+y2−2y−4=0$ and graph the circle.

Answer to Problem 10CR

Explanation of Solution

Chapter 4 Solutions

Additional Math Textbook Solutions