Algebra Examples

$25 x^{2} + y^{2} - 100 x - 2 y + 76 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $76$ from both sides of the equation.

$25 x^{2} + y^{2} - 100 x - 2 y = - 76$

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 $25 x^{2} + y^{2} - 100 x - 2 y = - 76$ .

$25 {(x - 2)}^{2} - 100 + y^{2} - 2 y = - 76$

Step 1.4

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

$25 {(x - 2)}^{2} + y^{2} - 2 y = - 76 + 100$

Step 1.5

Complete the square for $y^{2} - 2 y$ .

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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 $c$ .

$a = 1$

$b = - 2$

$c = 0$

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

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

Step 1.5.3.2

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

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Step 1.5.3.2.1

Factor $2$ out of $- 2$ .

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

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 $2 \cdot 1$ .

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

Step 1.5.3.2.2.2

Cancel the common factor.

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

Step 1.5.3.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

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

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

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

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

Step 1.5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.5.4.2.1.3

Cancel the common factor of $4$ .

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Step 1.5.4.2.1.3.1

Cancel the common factor.

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

Step 1.5.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 1.5.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 1.5.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 1.5.5

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

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

Step 1.6

Substitute ${(y - 1)}^{2} - 1$ for $y^{2} - 2 y$ in the equation $25 x^{2} + y^{2} - 100 x - 2 y = - 76$ .

$25 {(x - 2)}^{2} + {(y - 1)}^{2} - 1 = - 76 + 100$

Step 1.7

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

$25 {(x - 2)}^{2} + {(y - 1)}^{2} = - 76 + 100 + 1$

Step 1.8

Simplify $- 76 + 100 + 1$ .

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Step 1.8.1

Add $- 76$ and $100$ .

$25 {(x - 2)}^{2} + {(y - 1)}^{2} = 24 + 1$

Step 1.8.2

Add $24$ and $1$ .

$25 {(x - 2)}^{2} + {(y - 1)}^{2} = 25$

Step 1.9

Divide each term by $25$ to make the right side equal to one.

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

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 $1$ .

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

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.

$a = 5$

$b = 1$

$k = 1$

$h = 2$

Step 4

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

$(2, 1)$

Step 5