Precalculus Examples

$9 x^{2} + 25 y^{2} - 90 x + 150 y + 225 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $225$ from both sides of the equation.

$9 x^{2} + 25 y^{2} - 90 x + 150 y = - 225$

Step 1.2

Complete the square for $9 x^{2} - 90 x$ .

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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 $c$ .

$a = 9$

$b = - 90$

$c = 0$

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

$d = \frac{- 90}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 90$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 90$ .

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

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 $2 \cdot 9$ .

$d = \frac{2 \cdot - 45}{2 (9)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 45}{9}$

Step 1.2.3.2.2

Cancel the common factor of $- 45$ and $9$ .

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Step 1.2.3.2.2.1

Factor $9$ out of $- 45$ .

$d = \frac{9 \cdot - 5}{9}$

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 $9$ .

$d = \frac{9 \cdot - 5}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 5}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 0 - \frac{{(- 90)}^{2}}{4 \cdot 9}$

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 $- 90$ to the power of $2$ .

$e = 0 - \frac{8100}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{8100}{36}$

Step 1.2.4.2.1.3

Divide $8100$ by $36$ .

$e = 0 - 1 \cdot 225$

Step 1.2.4.2.1.4

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Step 1.2.4.2.2

Subtract $225$ from $0$ .

$e = - 225$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 5)}^{2} - 225$ .

$9 {(x - 5)}^{2} - 225$

Step 1.3

Substitute $9 {(x - 5)}^{2} - 225$ for $9 x^{2} - 90 x$ in the equation $9 x^{2} + 25 y^{2} - 90 x + 150 y = - 225$ .

$9 {(x - 5)}^{2} - 225 + 25 y^{2} + 150 y = - 225$

Step 1.4

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$9 {(x - 5)}^{2} + 25 y^{2} + 150 y = - 225 + 225$

Step 1.5

Complete the square for $25 y^{2} + 150 y$ .

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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 $c$ .

$a = 25$

$b = 150$

$c = 0$

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

$d = \frac{150}{2 \cdot 25}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $150$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $150$ .

$d = \frac{2 \cdot 75}{2 \cdot 25}$

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 $2 \cdot 25$ .

$d = \frac{2 \cdot 75}{2 (25)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{75}{25}$

Step 1.5.3.2.2

Cancel the common factor of $75$ and $25$ .

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Step 1.5.3.2.2.1

Factor $25$ out of $75$ .

$d = \frac{25 \cdot 3}{25}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 3}{25 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{25 \cdot 3}{25 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

$d = \frac{3}{1}$

Step 1.5.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{150^{2}}{4 \cdot 25}$

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 $150$ to the power of $2$ .

$e = 0 - \frac{22500}{4 \cdot 25}$

Step 1.5.4.2.1.2

Multiply $4$ by $25$ .

$e = 0 - \frac{22500}{100}$

Step 1.5.4.2.1.3

Divide $22500$ by $100$ .

$e = 0 - 1 \cdot 225$

Step 1.5.4.2.1.4

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Step 1.5.4.2.2

Subtract $225$ from $0$ .

$e = - 225$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 3)}^{2} - 225$ .

$25 {(y + 3)}^{2} - 225$

Step 1.6

Substitute $25 {(y + 3)}^{2} - 225$ for $25 y^{2} + 150 y$ in the equation $9 x^{2} + 25 y^{2} - 90 x + 150 y = - 225$ .

$9 {(x - 5)}^{2} + 25 {(y + 3)}^{2} - 225 = - 225 + 225$

Step 1.7

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$9 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = - 225 + 225 + 225$

Step 1.8

Simplify $- 225 + 225 + 225$ .

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Step 1.8.1

Add $- 225$ and $225$ .

$9 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = 0 + 225$

Step 1.8.2

Add $0$ and $225$ .

$9 {(x - 5)}^{2} + 25 {(y + 3)}^{2} = 225$

Step 1.9

Divide each term by $225$ to make the right side equal to one.

$\frac{9 {(x - 5)}^{2}}{225} + \frac{25 {(y + 3)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{25} + \frac{{(y + 3)}^{2}}{9} = 1$

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.

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

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.

$a = 5$

$b = 3$

$k = - 3$

$h = 5$

Step 4

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

$(5, - 3)$

Step 5