Find the Center and Radius (x^2-10x+25)+(y^2-16y+64)=16

$(x^{2} - 10 x + 25) + (y^{2} - 16 y + 64) = 16$

Step 1

Move all terms not containing a variable to the right side of the equation.

Subtract $89$ from both sides of the equation.

$x^{2} - 10 x + y^{2} - 16 y = 16 - 89$

Subtract $89$ from $16$ .

$x^{2} - 10 x + y^{2} - 16 y = - 73$

Step 2

Complete the square for $x^{2} - 10 x$ .

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

$a = 1$

$b = - 10$

$c = 0$

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

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

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Subtract $25$ from $0$ .

$e = - 25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 5)}^{2} - 25$ .

${(x - 5)}^{2} - 25$

Step 3

Substitute ${(x - 5)}^{2} - 25$ for $x^{2} - 10 x$ in the equation $x^{2} - 10 x + y^{2} - 16 y = - 73$ .

${(x - 5)}^{2} - 25 + y^{2} - 16 y = - 73$

Step 4

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x - 5)}^{2} + y^{2} - 16 y = - 73 + 25$

Step 5

Complete the square for $y^{2} - 16 y$ .

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

$a = 1$

$b = - 16$

$c = 0$

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

$d = \frac{- 16}{2 \cdot 1}$

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16$ .

$d = \frac{2 \cdot - 8}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 8}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 8}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 8}{1}$

Divide $- 8$ by $1$ .

$d = - 8$

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

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 16$ to the power of $2$ .

$e = 0 - \frac{256}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{256}{4}$

Divide $256$ by $4$ .

$e = 0 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Subtract $64$ from $0$ .

$e = - 64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 8)}^{2} - 64$ .

${(y - 8)}^{2} - 64$

Step 6

Substitute ${(y - 8)}^{2} - 64$ for $y^{2} - 16 y$ in the equation $x^{2} - 10 x + y^{2} - 16 y = - 73$ .

${(x - 5)}^{2} + {(y - 8)}^{2} - 64 = - 73 + 25$

Step 7

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

${(x - 5)}^{2} + {(y - 8)}^{2} = - 73 + 25 + 64$

Step 8

Simplify $- 73 + 25 + 64$ .

Add $- 73$ and $25$ .

${(x - 5)}^{2} + {(y - 8)}^{2} = - 48 + 64$

Add $- 48$ and $64$ .

${(x - 5)}^{2} + {(y - 8)}^{2} = 16$

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.

$r = 4$

$h = 5$

$k = 8$

Step 11

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

Center: $(5, 8)$

Step 12

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

Center: $(5, 8)$

Radius: $4$

Step 13

Find the Center and Radius (x^2-10x+25)+(y^2-16y+64)=16

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