Precalculus Examples

$9 x^{2} + 4 y^{2} - 18 x + 8 y - 23 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $23$ to both sides of the equation.

$9 x^{2} + 4 y^{2} - 18 x + 8 y = 23$

Step 1.2

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

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Step 1.2.1

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

$a = 9$

$b = - 18$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 18$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

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

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

Factor $9$ out of $- 9$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

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

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

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

Step 1.2.4.2.1.3

Divide $324$ by $36$ .

$e = 0 - 1 \cdot 9$

Step 1.2.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Step 1.2.4.2.2

Subtract $9$ from $0$ .

$e = - 9$

Step 1.2.5

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

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

Step 1.3

Substitute $9 {(x - 1)}^{2} - 9$ for $9 x^{2} - 18 x$ in the equation $9 x^{2} + 4 y^{2} - 18 x + 8 y = 23$ .

$9 {(x - 1)}^{2} - 9 + 4 y^{2} + 8 y = 23$

Step 1.4

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

$9 {(x - 1)}^{2} + 4 y^{2} + 8 y = 23 + 9$

Step 1.5

Complete the square for $4 y^{2} + 8 y$ .

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Step 1.5.1

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

$a = 4$

$b = 8$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $8$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $8$ .

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

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

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

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

Step 1.5.3.2.1.2.2

Cancel the common factor.

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

Step 1.5.3.2.1.2.3

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of $4$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

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

Step 1.5.3.2.2.2

Rewrite the expression.

$d = 1$

Step 1.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{8^{2}}{4 \cdot 4}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Step 1.5.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 1.5.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.5.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y + 1)}^{2} - 4$ .

$4 {(y + 1)}^{2} - 4$

Step 1.6

Substitute $4 {(y + 1)}^{2} - 4$ for $4 y^{2} + 8 y$ in the equation $9 x^{2} + 4 y^{2} - 18 x + 8 y = 23$ .

$9 {(x - 1)}^{2} + 4 {(y + 1)}^{2} - 4 = 23 + 9$

Step 1.7

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

$9 {(x - 1)}^{2} + 4 {(y + 1)}^{2} = 23 + 9 + 4$

Step 1.8

Simplify $23 + 9 + 4$ .

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Step 1.8.1

Add $23$ and $9$ .

$9 {(x - 1)}^{2} + 4 {(y + 1)}^{2} = 32 + 4$

Step 1.8.2

Add $32$ and $4$ .

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

Step 1.9

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

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

Step 1.10

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 $1$ .

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

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.

$a = 3$

$b = 2$

$k = - 1$

$h = 1$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(1, - 1)$

Step 5