BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.8, Problem 104E
To determine

To show: The equation x2+y2+6y+2=0 represents a circle and evaluate the center and radius of the circle.

Expert Solution

Answer to Problem 104E

The equation x2+y2+6y+2=0 represents a circle. The center of the circle is (0,3) and radius is 7 units.

Explanation of Solution

Given information:

The equation x2+y2+6y+2=0 .

Formula used:

In order to solve a quadratic equation, completing the square method is used that transforms the equation in the form of square trinomial.

Step1: Divide the equation by coefficient of x2 if it is not equal to 1.

Step 2: Take the square of the half of the coefficient of x and add it to both sides of the equation.

Step 3: Factor the equation.

The standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Calculation:

Consider the equation x2+y2+6y+2=0 .

Recall that in order to solve a quadratic equation, completing the square method is used that transforms the equation in the form of square trinomial.

Step1: Divide the equation by coefficient of x2 if it is not equal to 1.

Step 2: Take the square of the half of the coefficient of x and add it to both sides of the equation.

Step 3: Factor the equation.

In the provided equation, add 9 to both the sides of the equation,

  x2+y2+6y+2+9=0+9

Group the terms,

  x2+y2+6y+2+9=0+9(x2)+(y2+6y+9)=92

Factor out the trinomial, recall that (a+b)2=a2+2ab+b2 and (ab)2=a22ab+b2 .

Apply it,

  x2+y2+6y+2+9=0+9(x2)+(y2+6y+9)=92(x)2+(y+3)2=7

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Convert the equation obtained above in standard form,

  x2+y2+6y+2+9=0+9(x2)+(y2+6y+9)=92(x)2+(y+3)2=7(x0)2+(y(3))2=(7)2

Compare, (xh)2+(yk)2=r2 and (x0)2+(y(3))2=(7)2 .

Here, h=0,k=3 and r=7 .

Therefore, center of circle is (0,3) and radius is 7 .

Thus, the equation x2+y2+6y+2=0 represents a circle. The center of the circle is (0,3) and radius is 7 units.

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