Find the Foci 9x^2+4y^2-18x+16y-11=0

$9 x^{2} + 4 y^{2} - 18 x + 16 y - 11 = 0$

Step 1

Find the standard form of the ellipse.

Add $11$ to both sides of the equation.

$9 x^{2} + 4 y^{2} - 18 x + 16 y = 11$

Complete the square for $9 x^{2} - 18 x$ .

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

$a = 9$

$b = - 18$

$c = 0$

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

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

Simplify the right side.

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

Factor $2$ out of $- 18$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $9$ out of $- 9$ .

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

Cancel the common factors.

Factor $9$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{324}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{324}{36}$

Divide $324$ by $36$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Subtract $9$ from $0$ .

$e = - 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 1)}^{2} - 9$ .

$9 {(x - 1)}^{2} - 9$

Substitute $9 {(x - 1)}^{2} - 9$ for $9 x^{2} - 18 x$ in the equation $9 x^{2} + 4 y^{2} - 18 x + 16 y = 11$ .

$9 {(x - 1)}^{2} - 9 + 4 y^{2} + 16 y = 11$

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

$9 {(x - 1)}^{2} + 4 y^{2} + 16 y = 11 + 9$

Complete the square for $4 y^{2} + 16 y$ .

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

$a = 4$

$b = 16$

$c = 0$

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

$d = \frac{16}{2 \cdot 4}$

Simplify the right side.

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

Factor $2$ out of $16$ .

$d = \frac{2 \cdot 8}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 8}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot 8}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{8}{4}$

Cancel the common factor of $8$ and $4$ .

Factor $4$ out of $8$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot 2}{4 (1)}$

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $4$ .

$e = 0 - \frac{256}{16}$

Divide $256$ by $16$ .

$e = 0 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Subtract $16$ from $0$ .

$e = - 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y + 2)}^{2} - 16$ .

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

Substitute $4 {(y + 2)}^{2} - 16$ for $4 y^{2} + 16 y$ in the equation $9 x^{2} + 4 y^{2} - 18 x + 16 y = 11$ .

$9 {(x - 1)}^{2} + 4 {(y + 2)}^{2} - 16 = 11 + 9$

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$9 {(x - 1)}^{2} + 4 {(y + 2)}^{2} = 11 + 9 + 16$

Simplify $11 + 9 + 16$ .

Add $11$ and $9$ .

$9 {(x - 1)}^{2} + 4 {(y + 2)}^{2} = 20 + 16$

Add $20$ and $16$ .

$9 {(x - 1)}^{2} + 4 {(y + 2)}^{2} = 36$

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

$\frac{9 {(x - 1)}^{2}}{36} + \frac{4 {(y + 2)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{4} + \frac{{(y + 2)}^{2}}{9} = 1$

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.

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

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.

$a = 3$

$b = 2$

$k = - 2$

$h = 1$

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.

$\sqrt{{(3)}^{2} - {(2)}^{2}}$

Simplify.

Raise $3$ to the power of $2$ .

$\sqrt{9 - {(2)}^{2}}$

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

$\sqrt{9 - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

$\sqrt{9 - 4}$

Subtract $4$ from $9$ .

$\sqrt{5}$

Step 5

Find the foci.

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

$(h, k + c)$

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

$(1, - 2 + \sqrt{5})$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

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

$(1, - 2 - (\sqrt{5}))$

Simplify.

$(1, - 2 - \sqrt{5})$

Ellipses have two foci.

${Focus}_{1}$ : $(1, - 2 + \sqrt{5})$

${Focus}_{2}$ : $(1, - 2 - \sqrt{5})$

${Focus}_{1}$ : $(1, - 2 + \sqrt{5})$

${Focus}_{2}$ : $(1, - 2 - \sqrt{5})$

Step 6

Find the Foci 9x^2+4y^2-18x+16y-11=0

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