Calculus Examples

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

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $4$ from both sides of the equation.

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

Step 1.2

Complete the square for $4 x^{2} - 8 x$ .

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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 $c$ .

$a = 4$

$b = - 8$

$c = 0$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 8$ .

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

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 $2 \cdot 4$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

Factor $4$ out of $- 4$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

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

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

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

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

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

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

Step 1.2.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.2.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.2.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.2.5

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

$4 {(x - 1)}^{2} - 4$

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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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 $c$ .

$a = 1$

$b = 4$

$c = 0$

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

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

Step 1.5.3.2

Cancel the common factor of $4$ and $2$ .

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Step 1.5.3.2.1

Factor $2$ out of $4$ .

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

Step 1.5.3.2.2

Cancel the common factors.

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Step 1.5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 1.5.3.2.2.2

Cancel the common factor.

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

Step 1.5.3.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

$e = 0 - \frac{4^{2}}{4 \cdot 1}$

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.5.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.5.4.2.1.1.2

Cancel the common factors.

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Step 1.5.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 1$ .

$e = 0 - \frac{4 \cdot 4}{4 (1)}$

Step 1.5.4.2.1.1.2.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.5.4.2.1.1.2.3

Rewrite the expression.

$e = 0 - \frac{4}{1}$

Step 1.5.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 1.5.4.2.1.2

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

Simplify $- 4 + 4 + 4$ .

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Step 1.8.1

Add $- 4$ and $4$ .

$4 {(x - 1)}^{2} + {(y + 2)}^{2} = 0 + 4$

Step 1.8.2

Add $0$ and $4$ .

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

Step 1.9

Divide each term by $4$ to make the right side equal to one.

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

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 $1$ .

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

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.

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

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.

$a = 2$

$b = 1$

$k = - 2$

$h = 1$

Step 4

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

$(1, - 2)$

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.

$\sqrt{{(2)}^{2} - {(1)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\sqrt{4 - {(1)}^{2}}$

Step 5.3.2

One to any power is one.

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

Step 5.3.3

Multiply $- 1$ by $1$ .

$\sqrt{4 - 1}$

Step 5.3.4

Subtract $1$ from $4$ .

$\sqrt{3}$

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.

$(1, - 2 + 2)$

Step 6.3

Simplify.

$(1, 0)$

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.

$(1, - 2 - (2))$

Step 6.6

Simplify.

$(1, - 4)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 4)$

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 4)$

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.

$(1, - 2 + \sqrt{3})$

Step 7.3

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

$(h, k - c)$

Step 7.4

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

$(1, - 2 - (\sqrt{3}))$

Step 7.5

Simplify.

$(1, - 2 - \sqrt{3})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(1, - 2 + \sqrt{3})$

${Focus}_{2}$ : $(1, - 2 - \sqrt{3})$

${Focus}_{1}$ : $(1, - 2 + \sqrt{3})$

${Focus}_{2}$ : $(1, - 2 - \sqrt{3})$

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.

$\frac{\sqrt{{(2)}^{2} - {(1)}^{2}}}{2}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

$\frac{\sqrt{4 - 1^{2}}}{2}$

Step 8.3.2

One to any power is one.

$\frac{\sqrt{4 - 1 \cdot 1}}{2}$

Step 8.3.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{4 - 1}}{2}$

Step 8.3.4

Subtract $1$ from $4$ .

$\frac{\sqrt{3}}{2}$

Step 9

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

Center: $(1, - 2)$

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 4)$

${Focus}_{1}$ : $(1, - 2 + \sqrt{3})$

${Focus}_{2}$ : $(1, - 2 - \sqrt{3})$

Eccentricity: $\frac{\sqrt{3}}{2}$

Step 10