Find the Center and Radius x^2+y^2-18x+20y+177=0

$x^{2} + y^{2} - 18 x + 20 y + 177 = 0$

Step 1

Subtract $177$ from both sides of the equation.

$x^{2} + y^{2} - 18 x + 20 y = - 177$

Step 2

Complete the square for $x^{2} - 18 x$ .

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

$a = 1$

$b = - 18$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 18}{2 \cdot 1}$

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 9}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 9}{1}$

Divide $- 9$ by $1$ .

$d = - 9$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{324}{4}$

Divide $324$ by $4$ .

$e = 0 - 1 \cdot 81$

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Subtract $81$ from $0$ .

$e = - 81$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 9)}^{2} - 81$ .

${(x - 9)}^{2} - 81$

Step 3

Substitute ${(x - 9)}^{2} - 81$ for $x^{2} - 18 x$ in the equation $x^{2} + y^{2} - 18 x + 20 y = - 177$ .

${(x - 9)}^{2} - 81 + y^{2} + 20 y = - 177$

Step 4

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

${(x - 9)}^{2} + y^{2} + 20 y = - 177 + 81$

Step 5

Complete the square for $y^{2} + 20 y$ .

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

$a = 1$

$b = 20$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{20}{2 \cdot 1}$

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot 10}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{10}{1}$

Divide $10$ by $1$ .

$d = 10$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{20^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 0 - \frac{400}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{400}{4}$

Divide $400$ by $4$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 10)}^{2} - 100$ .

${(y + 10)}^{2} - 100$

Step 6

Substitute ${(y + 10)}^{2} - 100$ for $y^{2} + 20 y$ in the equation $x^{2} + y^{2} - 18 x + 20 y = - 177$ .

${(x - 9)}^{2} + {(y + 10)}^{2} - 100 = - 177 + 81$

Step 7

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

${(x - 9)}^{2} + {(y + 10)}^{2} = - 177 + 81 + 100$

Step 8

Simplify $- 177 + 81 + 100$ .

Add $- 177$ and $81$ .

${(x - 9)}^{2} + {(y + 10)}^{2} = - 96 + 100$

Add $- 96$ and $100$ .

${(x - 9)}^{2} + {(y + 10)}^{2} = 4$

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.

$r = 2$

$h = 9$

$k = - 10$

Step 11

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

Center: $(9, - 10)$

Step 12

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

Center: $(9, - 10)$

Radius: $2$

Step 13

Find the Center and Radius x^2+y^2-18x+20y+177=0

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