Find the Foci x^2+9y^2-14x+36y+49=0

$x^{2} + 9 y^{2} - 14 x + 36 y + 49 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $49$ from both sides of the equation.

$x^{2} + 9 y^{2} - 14 x + 36 y = - 49$

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

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

Substitute ${(x - 7)}^{2} - 49$ for $x^{2} - 14 x$ in the equation $x^{2} + 9 y^{2} - 14 x + 36 y = - 49$ .

${(x - 7)}^{2} - 49 + 9 y^{2} + 36 y = - 49$

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

${(x - 7)}^{2} + 9 y^{2} + 36 y = - 49 + 49$

Complete the square for $9 y^{2} + 36 y$ .

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

$a = 9$

$b = 36$

$c = 0$

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

$d = \frac{36}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$d = \frac{2 \cdot 18}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot 18}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot 18}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{18}{9}$

Cancel the common factor of $18$ and $9$ .

Factor $9$ out of $18$ .

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

Cancel the common factors.

Factor $9$ out of $9$ .

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

Cancel the common factor.

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

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

$e = 0 - \frac{36^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{1296}{36}$

Divide $1296$ by $36$ .

$e = 0 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Subtract $36$ from $0$ .

$e = - 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y + 2)}^{2} - 36$ .

$9 {(y + 2)}^{2} - 36$

Substitute $9 {(y + 2)}^{2} - 36$ for $9 y^{2} + 36 y$ in the equation $x^{2} + 9 y^{2} - 14 x + 36 y = - 49$ .

${(x - 7)}^{2} + 9 {(y + 2)}^{2} - 36 = - 49 + 49$

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

${(x - 7)}^{2} + 9 {(y + 2)}^{2} = - 49 + 49 + 36$

Simplify $- 49 + 49 + 36$ .

Add $- 49$ and $49$ .

${(x - 7)}^{2} + 9 {(y + 2)}^{2} = 0 + 36$

Add $0$ and $36$ .

${(x - 7)}^{2} + 9 {(y + 2)}^{2} = 36$

Divide each term by $36$ to make the right side equal to one.

$\frac{{(x - 7)}^{2}}{36} + \frac{9 {(y + 2)}^{2}}{36} = \frac{36}{36}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(x - 7)}^{2}}{36} + \frac{{(y + 2)}^{2}}{4} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 6$

$b = 2$

$k = - 2$

$h = 7$

Step 4

Find $c$ , the distance from the center to a focus.

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(6)}^{2} - {(2)}^{2}}$

Simplify.

Raise $6$ to the power of $2$ .

$\sqrt{36 - {(2)}^{2}}$

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

$\sqrt{36 - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

$\sqrt{36 - 4}$

Subtract $4$ from $36$ .

$\sqrt{32}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$\sqrt{16 (2)}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$4 \sqrt{2}$

Step 5

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(7 + 4 \sqrt{2}, - 2)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(7 - (4 \sqrt{2}), - 2)$

Simplify.

$(7 - 4 \sqrt{2}, - 2)$

Ellipses have two foci.

${Focus}_{1}$ : $(7 + 4 \sqrt{2}, - 2)$

${Focus}_{2}$ : $(7 - 4 \sqrt{2}, - 2)$

${Focus}_{1}$ : $(7 + 4 \sqrt{2}, - 2)$

${Focus}_{2}$ : $(7 - 4 \sqrt{2}, - 2)$

Step 6

Find the Foci x^2+9y^2-14x+36y+49=0

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