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 106E
To determine

To show: The equation x2+y2+12x+2y+116=0 represents a circle and evaluate the center and radius of the circle.

Expert Solution

Answer to Problem 106E

The equation x2+y2+12x+2y+116=0 represents a circle. The center of the circle is (14,1) and radius is 1 units.

Explanation of Solution

Given information:

The equation x2+y2+12x+2y+116=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+12x+2y+116=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 116 to both the sides of the equation,

  x2+y2+12x+2y+116+116=116

Again add 1 to both the sides of the equation,

  x2+y2+12x+2y+116+116+1=116+1

Group the terms,

  x2+y2+12x+2y+116+116+1=116+1(x2+12x+116)+(y2+2y+1)=1

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

Apply it,

  x2+y2+12x+2y+116+116+1=116+1(x2+12x+116)+(y2+2y+1)=1(x+14)2+(y+1)2=1

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+12x+2y+116+116+1=116+1(x2+12x+116)+(y2+2y+1)=1(x+14)2+(y+1)2=1(x(14))2+(y(1))2=(1)2

Compare, (xh)2+(yk)2=r2 and (x(14))2+(y(1))2=(1)2 .

Here, h=14,k=1 and r=1 .

Therefore, center of circle is (14,1) and radius is 1 .

Thus, the equation x2+y2+12x+2y+116=0 represents a circle. The center of the circle is (14,1) and radius is 1 units.

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