Find the Foci 11x^2-2y^2+66x-16y+45=0

$11 x^{2} - 2 y^{2} + 66 x - 16 y + 45 = 0$

Step 1

Find the standard form of the hyperbola.

Subtract $45$ from both sides of the equation.

$11 x^{2} - 2 y^{2} + 66 x - 16 y = - 45$

Complete the square for $11 x^{2} + 66 x$ .

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

$a = 11$

$b = 66$

$c = 0$

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

$d = \frac{66}{2 \cdot 11}$

Simplify the right side.

Cancel the common factor of $66$ and $2$ .

Factor $2$ out of $66$ .

$d = \frac{2 \cdot 33}{2 \cdot 11}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 11$ .

$d = \frac{2 \cdot 33}{2 (11)}$

Cancel the common factor.

$d = \frac{2 \cdot 33}{2 \cdot 11}$

Rewrite the expression.

$d = \frac{33}{11}$

Cancel the common factor of $33$ and $11$ .

Factor $11$ out of $33$ .

$d = \frac{11 \cdot 3}{11}$

Cancel the common factors.

Factor $11$ out of $11$ .

$d = \frac{11 \cdot 3}{11 (1)}$

Cancel the common factor.

$d = \frac{11 \cdot 3}{11 \cdot 1}$

Rewrite the expression.

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

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{66^{2}}{4 \cdot 11}$

Simplify the right side.

Simplify each term.

Raise $66$ to the power of $2$ .

$e = 0 - \frac{4356}{4 \cdot 11}$

Multiply $4$ by $11$ .

$e = 0 - \frac{4356}{44}$

Divide $4356$ by $44$ .

$e = 0 - 1 \cdot 99$

Multiply $- 1$ by $99$ .

$e = 0 - 99$

Subtract $99$ from $0$ .

$e = - 99$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $11 {(x + 3)}^{2} - 99$ .

$11 {(x + 3)}^{2} - 99$

Substitute $11 {(x + 3)}^{2} - 99$ for $11 x^{2} + 66 x$ in the equation $11 x^{2} - 2 y^{2} + 66 x - 16 y = - 45$ .

$11 {(x + 3)}^{2} - 99 - 2 y^{2} - 16 y = - 45$

Move $- 99$ to the right side of the equation by adding $99$ to both sides.

$11 {(x + 3)}^{2} - 2 y^{2} - 16 y = - 45 + 99$

Complete the square for $- 2 y^{2} - 16 y$ .

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

$a = - 2$

$b = - 16$

$c = 0$

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

$d = \frac{- 16}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $- 16$ and $2$ .

Factor $2$ out of $- 16$ .

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

Cancel the common factors.

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

$d = \frac{2 \cdot - 8}{2 (- 2)}$

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $- 2$ out of $- 8$ .

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

Cancel the common factors.

Factor $- 2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$d = 4$

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

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $- 16$ to the power of $2$ .

$e = 0 - \frac{256}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = 0 - \frac{256}{- 8}$

Divide $256$ by $- 8$ .

$e = 0 - - 32$

Multiply $- 1$ by $- 32$ .

$e = 0 + 32$

Add $0$ and $32$ .

$e = 32$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(y + 4)}^{2} + 32$ .

$- 2 {(y + 4)}^{2} + 32$

Substitute $- 2 {(y + 4)}^{2} + 32$ for $- 2 y^{2} - 16 y$ in the equation $11 x^{2} - 2 y^{2} + 66 x - 16 y = - 45$ .

$11 {(x + 3)}^{2} - 2 {(y + 4)}^{2} + 32 = - 45 + 99$

Move $32$ to the right side of the equation by adding $32$ to both sides.

$11 {(x + 3)}^{2} - 2 {(y + 4)}^{2} = - 45 + 99 - 32$

Simplify $- 45 + 99 - 32$ .

Add $- 45$ and $99$ .

$11 {(x + 3)}^{2} - 2 {(y + 4)}^{2} = 54 - 32$

Subtract $32$ from $54$ .

$11 {(x + 3)}^{2} - 2 {(y + 4)}^{2} = 22$

Divide each term by $22$ to make the right side equal to one.

$\frac{11 {(x + 3)}^{2}}{22} - \frac{2 {(y + 4)}^{2}}{22} = \frac{22}{22}$

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 $1$ .

$\frac{{(x + 3)}^{2}}{2} - \frac{{(y + 4)}^{2}}{11} = 1$

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.

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

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, $a$ .

$a = \sqrt{2}$

$b = \sqrt{11}$

$k = - 4$

$h = - 3$

Step 4

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.

$\sqrt{{(\sqrt{2})}^{2} + {(\sqrt{11})}^{2}}$

Simplify.

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

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

$\sqrt{{(2^{\frac{1}{2}})}^{2} + {(\sqrt{11})}^{2}}$

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

$\sqrt{2^{\frac{1}{2} \cdot 2} + {(\sqrt{11})}^{2}}$

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

$\sqrt{2^{\frac{2}{2}} + {(\sqrt{11})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{2^{\frac{2}{2}} + {(\sqrt{11})}^{2}}$

Rewrite the expression.

$\sqrt{2^{1} + {(\sqrt{11})}^{2}}$

Evaluate the exponent.

$\sqrt{2 + {(\sqrt{11})}^{2}}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{2 + {(11^{\frac{1}{2}})}^{2}}$

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

$\sqrt{2 + 11^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{2 + 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{2 + 11^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{2 + 11^{1}}$

Evaluate the exponent.

$\sqrt{2 + 11}$

Add $2$ and $11$ .

$\sqrt{13}$

Step 5

Find the foci.

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

$(h + c, k)$

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

$(- 3 + \sqrt{13}, - 4)$

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

$(h - c, k)$

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

$(- 3 - \sqrt{13}, - 4)$

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

$(- 3 + \sqrt{13}, - 4), (- 3 - \sqrt{13}, - 4)$

Step 6

Find the Foci 11x^2-2y^2+66x-16y+45=0

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