Precalculus Examples

$25 x^{2} + 9 y^{2} - 100 x + 54 y - 44 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Add $44$ to both sides of the equation.

$25 x^{2} + 9 y^{2} - 100 x + 54 y = 44$

Step 1.2

Complete the square for $25 x^{2} - 100 x$ .

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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 $c$ .

$a = 25$

$b = - 100$

$c = 0$

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

$d = \frac{- 100}{2 \cdot 25}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 100$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 100$ .

$d = \frac{2 \cdot - 50}{2 \cdot 25}$

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 $2 \cdot 25$ .

$d = \frac{2 \cdot - 50}{2 (25)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 50}{2 \cdot 25}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 50}{25}$

Step 1.2.3.2.2

Cancel the common factor of $- 50$ and $25$ .

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Step 1.2.3.2.2.1

Factor $25$ out of $- 50$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $25$ out of $25$ .

$d = \frac{25 \cdot - 2}{25 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{25 \cdot - 2}{25 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 0 - \frac{{(- 100)}^{2}}{4 \cdot 25}$

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 $- 100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot 25}$

Step 1.2.4.2.1.2

Multiply $4$ by $25$ .

$e = 0 - \frac{10000}{100}$

Step 1.2.4.2.1.3

Divide $10000$ by $100$ .

$e = 0 - 1 \cdot 100$

Step 1.2.4.2.1.4

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Step 1.2.4.2.2

Subtract $100$ from $0$ .

$e = - 100$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(x - 2)}^{2} - 100$ .

$25 {(x - 2)}^{2} - 100$

Step 1.3

Substitute $25 {(x - 2)}^{2} - 100$ for $25 x^{2} - 100 x$ in the equation $25 x^{2} + 9 y^{2} - 100 x + 54 y = 44$ .

$25 {(x - 2)}^{2} - 100 + 9 y^{2} + 54 y = 44$

Step 1.4

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

$25 {(x - 2)}^{2} + 9 y^{2} + 54 y = 44 + 100$

Step 1.5

Complete the square for $9 y^{2} + 54 y$ .

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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 $c$ .

$a = 9$

$b = 54$

$c = 0$

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

$d = \frac{54}{2 \cdot 9}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $54$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $54$ .

$d = \frac{2 \cdot 27}{2 \cdot 9}$

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 $2 \cdot 9$ .

$d = \frac{2 \cdot 27}{2 (9)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 27}{2 \cdot 9}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{27}{9}$

Step 1.5.3.2.2

Cancel the common factor of $27$ and $9$ .

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Step 1.5.3.2.2.1

Factor $9$ out of $27$ .

$d = \frac{9 \cdot 3}{9}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot 3}{9 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot 3}{9 \cdot 1}$

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

$e = 0 - \frac{54^{2}}{4 \cdot 9}$

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 $54$ to the power of $2$ .

$e = 0 - \frac{2916}{4 \cdot 9}$

Step 1.5.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{2916}{36}$

Step 1.5.4.2.1.3

Divide $2916$ by $36$ .

$e = 0 - 1 \cdot 81$

Step 1.5.4.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 1.5.4.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y + 3)}^{2} - 81$ .

$9 {(y + 3)}^{2} - 81$

Step 1.6

Substitute $9 {(y + 3)}^{2} - 81$ for $9 y^{2} + 54 y$ in the equation $25 x^{2} + 9 y^{2} - 100 x + 54 y = 44$ .

$25 {(x - 2)}^{2} + 9 {(y + 3)}^{2} - 81 = 44 + 100$

Step 1.7

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

$25 {(x - 2)}^{2} + 9 {(y + 3)}^{2} = 44 + 100 + 81$

Step 1.8

Simplify $44 + 100 + 81$ .

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Step 1.8.1

Add $44$ and $100$ .

$25 {(x - 2)}^{2} + 9 {(y + 3)}^{2} = 144 + 81$

Step 1.8.2

Add $144$ and $81$ .

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

Step 1.9

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

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

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 $1$ .

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

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.

$a = 5$

$b = 3$

$k = - 3$

$h = 2$

Step 4

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

$(2, - 3)$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$\sqrt{{(5)}^{2} - {(3)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $5$ to the power of $2$ .

$\sqrt{25 - {(3)}^{2}}$

Step 5.3.2

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

$\sqrt{25 - 1 \cdot 9}$

Step 5.3.3

Multiply $- 1$ by $9$ .

$\sqrt{25 - 9}$

Step 5.3.4

Subtract $9$ from $25$ .

$\sqrt{16}$

Step 5.3.5

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

$\sqrt{4^{2}}$

Step 5.3.6

Pull terms out from under the radical, assuming positive real numbers.

$4$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

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

$(2, - 3 + 5)$

Step 6.3

Simplify.

$(2, 2)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.5

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

$(2, - 3 - (5))$

Step 6.6

Simplify.

$(2, - 8)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(2, 2)$

${Vertex}_{2}$ : $(2, - 8)$

${Vertex}_{1}$ : $(2, 2)$

${Vertex}_{2}$ : $(2, - 8)$

Step 7

Find the foci.

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Step 7.1

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

$(h, k + c)$

Step 7.2

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

$(2, - 3 + 4)$

Step 7.3

Simplify.

$(2, 1)$

Step 7.4

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

$(h, k - c)$

Step 7.5

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

$(2, - 3 - (4))$

Step 7.6

Simplify.

$(2, - 7)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(2, 1)$

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

${Focus}_{1}$ : $(2, 1)$

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

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

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

$\frac{\sqrt{{(5)}^{2} - {(3)}^{2}}}{5}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 - 3^{2}}}{5}$

Step 8.3.2

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

$\frac{\sqrt{25 - 1 \cdot 9}}{5}$

Step 8.3.3

Multiply $- 1$ by $9$ .

$\frac{\sqrt{25 - 9}}{5}$

Step 8.3.4

Subtract $9$ from $25$ .

$\frac{\sqrt{16}}{5}$

Step 8.3.5

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

$\frac{\sqrt{4^{2}}}{5}$

Step 8.3.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4}{5}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(2, - 3)$

${Vertex}_{1}$ : $(2, 2)$

${Vertex}_{2}$ : $(2, - 8)$

${Focus}_{1}$ : $(2, 1)$

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

Eccentricity: $\frac{4}{5}$

Step 10