Find the Properties y^2-2x^2-4x-4y=0

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

Step 1

Find the standard form of the hyperbola.

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

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

$a = 1$

$b = - 4$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

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

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2$ by $1$ .

$d = - 2$

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

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

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

Simplify the right side.

Simplify each term.

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $4^{2}$ by $1$ .

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

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

Substitute ${(y - 2)}^{2} - 4$ for $y^{2} - 4 y$ in the equation $y^{2} - 2 x^{2} - 4 x - 4 y = 0$ .

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

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

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

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

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

$a = - 2$

$b = - 4$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

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

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

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

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

Simplify the right side.

Simplify each term.

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $4^{2}$ by $1$ .

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$e = 0 - \frac{4}{- 2}$

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

Factor $2$ out of $4$ .

$e = 0 - \frac{2 (2)}{- 2}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$e = 0 - (- 1 \cdot 2)$

Multiply $- (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$e = 0 - - 2$

Multiply $- 1$ by $- 2$ .

$e = 0 + 2$

Add $0$ and $2$ .

$e = 2$

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

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

Substitute $- 2 {(x + 1)}^{2} + 2$ for $- 2 x^{2} - 4 x$ in the equation $y^{2} - 2 x^{2} - 4 x - 4 y = 0$ .

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

Move $2$ to the right side of the equation by adding $2$ to both sides.

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

Simplify $0 + 4 - 2$ .

Add $0$ and $4$ .

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

Subtract $2$ from $4$ .

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

Divide each term by $2$ to make the right side equal to one.

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

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 $1$ .

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

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = \sqrt{2}$

$b = 1$

$k = 2$

$h = - 1$

Step 4

The center of a hyperbola 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.

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

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

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

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

Simplify.

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{2^{\frac{1}{2} \cdot 2} + {(1)}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the expression.

One to any power is one.

$\sqrt{2 + 1}$

Add $2$ and $1$ .

$\sqrt{3}$

Step 6

Find the vertices.

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

$(h, k + a)$

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

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

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

$(h, k - a)$

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

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

The vertices of a hyperbola follow the form of $(h, k \pm a)$ . Hyperbolas have two vertices.

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

Step 7

Find the foci.

The first focus of a hyperbola can be found by adding $c$ to $k$ .

$(h, k + c)$

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

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

The second focus of a hyperbola can be found by subtracting $c$ from $k$ .

$(h, k - c)$

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

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

The foci of a hyperbola follow the form of $(h, k \pm \sqrt{a^{2} + b^{2}})$ . Hyperbolas have two foci.

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

Step 8

Find the eccentricity.

Find the eccentricity by using the following formula.

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

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

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

Simplify.

Simplify the numerator.

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2^{\frac{1}{2} \cdot 2} + 1^{2}}}{\sqrt{2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2^{\frac{2}{2}} + 1^{2}}}{\sqrt{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2^{\frac{2}{2}} + 1^{2}}}{\sqrt{2}}$

Rewrite the expression.

$\frac{\sqrt{2^{1} + 1^{2}}}{\sqrt{2}}$

Evaluate the exponent.

$\frac{\sqrt{2 + 1^{2}}}{\sqrt{2}}$

One to any power is one.

$\frac{\sqrt{2 + 1}}{\sqrt{2}}$

Add $2$ and $1$ .

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

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{2}$

Step 9

Find the focal parameter.

Find the value of the focal parameter of the hyperbola by using the following formula.

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

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{1^{2}}{\sqrt{3}}$

Simplify.

One to any power is one.

$\frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

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

Step 10

The asymptotes follow the form $y = \pm \frac{a (x - h)}{b} + k$ because this hyperbola opens up and down.

$y = \pm \sqrt{2} \cdot (x - (- 1)) + 2$

Step 11

Simplify to find the first asymptote.

Remove parentheses.

$y = \sqrt{2} \cdot (x - (- 1)) + 2$

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = \sqrt{2} \cdot (x + 1) + 2$

Apply the distributive property.

$y = \sqrt{2} x + \sqrt{2} \cdot 1 + 2$

Multiply $\sqrt{2}$ by $1$ .

$y = \sqrt{2} x + \sqrt{2} + 2$

Step 12

Simplify to find the second asymptote.

Remove parentheses.

$y = - \sqrt{2} \cdot (x - (- 1)) + 2$

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = - \sqrt{2} \cdot (x + 1) + 2$

Apply the distributive property.

$y = - \sqrt{2} x - \sqrt{2} \cdot 1 + 2$

Multiply $- 1$ by $1$ .

$y = - \sqrt{2} x - \sqrt{2} + 2$

Step 13

This hyperbola has two asymptotes.

$y = \sqrt{2} x + \sqrt{2} + 2, y = - \sqrt{2} x - \sqrt{2} + 2$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(- 1, 2)$

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

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

Eccentricity: $\frac{\sqrt{6}}{2}$

Focal Parameter: $\frac{\sqrt{3}}{3}$

Asymptotes: $y = \sqrt{2} x + \sqrt{2} + 2$ , $y = - \sqrt{2} x - \sqrt{2} + 2$

Step 15

Find the Properties y^2-2x^2-4x-4y=0

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