Find the Center and Radius x^2+y^2-14x+2y+41=0

$x^{2} + y^{2} - 14 x + 2 y + 41 = 0$

Step 1

Subtract $41$ from both sides of the equation.

$x^{2} + y^{2} - 14 x + 2 y = - 41$

Step 2

Complete the square for $x^{2} - 14 x$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 14$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 14}{2 \cdot 1}$

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 7}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 7}{1}$

Divide $- 7$ by $1$ .

$d = - 7$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 14)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 14$ to the power of $2$ .

$e = 0 - \frac{196}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{196}{4}$

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Subtract $49$ from $0$ .

$e = - 49$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 7)}^{2} - 49$ .

${(x - 7)}^{2} - 49$

Step 3

Substitute ${(x - 7)}^{2} - 49$ for $x^{2} - 14 x$ in the equation $x^{2} + y^{2} - 14 x + 2 y = - 41$ .

${(x - 7)}^{2} - 49 + y^{2} + 2 y = - 41$

Step 4

Move $- 49$ to the right side of the equation by adding $49$ to both sides.

${(x - 7)}^{2} + y^{2} + 2 y = - 41 + 49$

Step 5

Complete the square for $y^{2} + 2 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 2$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot 1}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot 1}$

Rewrite the expression.

$d = 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $2$ to the power of $2$ .

$e = 0 - \frac{4}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{4}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Subtract $1$ from $0$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 1)}^{2} - 1$ .

${(y + 1)}^{2} - 1$

Step 6

Substitute ${(y + 1)}^{2} - 1$ for $y^{2} + 2 y$ in the equation $x^{2} + y^{2} - 14 x + 2 y = - 41$ .

${(x - 7)}^{2} + {(y + 1)}^{2} - 1 = - 41 + 49$

Step 7

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x - 7)}^{2} + {(y + 1)}^{2} = - 41 + 49 + 1$

Step 8

Simplify $- 41 + 49 + 1$ .

Add $- 41$ and $49$ .

${(x - 7)}^{2} + {(y + 1)}^{2} = 8 + 1$

Add $8$ and $1$ .

${(x - 7)}^{2} + {(y + 1)}^{2} = 9$

Step 9

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 10

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = 3$

$h = 7$

$k = - 1$

Step 11

The center of the circle is found at $(h, k)$ .

Center: $(7, - 1)$

Step 12

These values represent the important values for graphing and analyzing a circle.

Center: $(7, - 1)$

Radius: $3$

Step 13

Find the Center and Radius x^2+y^2-14x+2y+41=0

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