Find the Center and Radius x^2+y^2+18x+14y+105=0

$x^{2} + y^{2} + 18 x + 14 y + 105 = 0$

Step 1

Subtract $105$ from both sides of the equation.

$x^{2} + y^{2} + 18 x + 14 y = - 105$

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 $x^{2} + y^{2} + 18 x + 14 y = - 105$ .

${(x + 9)}^{2} - 81 + y^{2} + 14 y = - 105$

Step 4

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

${(x + 9)}^{2} + y^{2} + 14 y = - 105 + 81$

Step 5

Complete the square for $y^{2} + 14 y$ .

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 ${(y + 7)}^{2} - 49$ .

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

Step 6

Substitute ${(y + 7)}^{2} - 49$ for $y^{2} + 14 y$ in the equation $x^{2} + y^{2} + 18 x + 14 y = - 105$ .

${(x + 9)}^{2} + {(y + 7)}^{2} - 49 = - 105 + 81$

Step 7

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

${(x + 9)}^{2} + {(y + 7)}^{2} = - 105 + 81 + 49$

Step 8

Simplify $- 105 + 81 + 49$ .

Add $- 105$ and $81$ .

${(x + 9)}^{2} + {(y + 7)}^{2} = - 24 + 49$

Add $- 24$ and $49$ .

${(x + 9)}^{2} + {(y + 7)}^{2} = 25$

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.

$r = 5$

$h = - 9$

$k = - 7$

Step 11

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

Center: $(- 9, - 7)$

Step 12

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

Center: $(- 9, - 7)$

Radius: $5$

Step 13

Find the Center and Radius x^2+y^2+18x+14y+105=0

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