# The equation x 2 + y 2 − 4 x + 10 y + 13 = 0 represents a circle and evaluate the center and radius of the circle.

### Precalculus: Mathematics for Calcu...

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

### Precalculus: Mathematics for Calcu...

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

#### Solutions

Chapter 1.8, Problem 103E
To determine

## To show: The equation x2+y2−4x+10y+13=0 represents a circle and evaluate the center and radius of the circle.

Expert Solution

The equation x2+y24x+10y+13=0 represents a circle. The center of the circle is (2,5) and radius is 4 units.

### Explanation of Solution

Given information:

The equation x2+y24x+10y+13=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+y24x+10y+13=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 4 to both the sides of the equation,

x2+y24x+10y+13+4=0+4

Next, add 25 to both the sides of the equation,

x2+y24x+10y+13+4+25=0+4+25

Group the terms,

x2+y24x+10y+13+4+25=0+4+25(x24x+4)+(y2+10y+25)=2913

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

Apply it,

x2+y24x+10y+13+4+25=0+4+25(x24x+4)+(y2+10y+25)=2913(x2)2+(y+5)2=16

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+y24x+10y+13+4+25=0+4+25(x24x+4)+(y2+10y+25)=2913(x2)2+(y+5)2=16(x2)2+(y(5))2=42

Compare, (xh)2+(yk)2=r2 and (x2)2+(y(5))2=42 .

Here, h=2,k=5 and r=4 .

Therefore, center of circle is (2,5) and radius is 4.

Thus, the equation x2+y24x+10y+13=0 represents a circle. The center of the circle is (2,5) and radius is 4 units.

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