Find the Center and Radius x^2+y^2-4x+16y+46=0

$x^{2} + y^{2} - 4 x + 16 y + 46 = 0$

Step 1

Subtract $46$ from both sides of the equation.

$x^{2} + y^{2} - 4 x + 16 y = - 46$

Step 2

Complete the square for $x^{2} - 4 x$ .

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

$a = 1$

$b = - 4$

$c = 0$

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

$d = \frac{- 4}{2 \cdot 1}$

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$d = - 2$

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 2)}^{2} - 4$ .

${(x - 2)}^{2} - 4$

Step 3

Substitute ${(x - 2)}^{2} - 4$ for $x^{2} - 4 x$ in the equation $x^{2} + y^{2} - 4 x + 16 y = - 46$ .

${(x - 2)}^{2} - 4 + y^{2} + 16 y = - 46$

Step 4

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x - 2)}^{2} + y^{2} + 16 y = - 46 + 4$

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 $x^{2} + y^{2} - 4 x + 16 y = - 46$ .

${(x - 2)}^{2} + {(y + 8)}^{2} - 64 = - 46 + 4$

Step 7

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

${(x - 2)}^{2} + {(y + 8)}^{2} = - 46 + 4 + 64$

Step 8

Simplify $- 46 + 4 + 64$ .

Add $- 46$ and $4$ .

${(x - 2)}^{2} + {(y + 8)}^{2} = - 42 + 64$

Add $- 42$ and $64$ .

${(x - 2)}^{2} + {(y + 8)}^{2} = 22$

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.

$r = \sqrt{22}$

$h = 2$

$k = - 8$

Step 11

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

Center: $(2, - 8)$

Step 12

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

Center: $(2, - 8)$

Radius: $\sqrt{22}$

Step 13

Find the Center and Radius x^2+y^2-4x+16y+46=0

Related Questions

Support LCMGCF.com by disabling your adblocker.