Find the Foci 9x^2-4y^2-72x=0

$9 x^{2} - 4 y^{2} - 72 x = 0$

Step 1

Find the standard form of the hyperbola.

Complete the square for $9 x^{2} - 72 x$ .

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

$a = 9$

$b = - 72$

$c = 0$

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

$d = \frac{- 72}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 72$ and $2$ .

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 36}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 36}{9}$

Cancel the common factor of $- 36$ and $9$ .

Factor $9$ out of $- 36$ .

$d = \frac{9 \cdot - 4}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 4}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 4}{9 \cdot 1}$

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

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{5184}{36}$

Divide $5184$ by $36$ .

$e = 0 - 1 \cdot 144$

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Subtract $144$ from $0$ .

$e = - 144$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 4)}^{2} - 144$ .

$9 {(x - 4)}^{2} - 144$

Substitute $9 {(x - 4)}^{2} - 144$ for $9 x^{2} - 72 x$ in the equation $9 x^{2} - 4 y^{2} - 72 x = 0$ .

$9 {(x - 4)}^{2} - 144 - 4 y^{2} = 0$

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

$9 {(x - 4)}^{2} - 4 y^{2} = 0 + 144$

Add $0$ and $144$ .

$9 {(x - 4)}^{2} - 4 y^{2} = 144$

Divide each term by $144$ to make the right side equal to one.

$\frac{9 {(x - 4)}^{2}}{144} - \frac{4 y^{2}}{144} = \frac{144}{144}$

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 $1$ .

$\frac{{(x - 4)}^{2}}{16} - \frac{y^{2}}{36} = 1$

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, $a$ .

$a = 4$

$b = 6$

$k = 0$

$h = 4$

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.

$\sqrt{{(4)}^{2} + {(6)}^{2}}$

Simplify.

Raise $4$ to the power of $2$ .

$\sqrt{16 + {(6)}^{2}}$

Raise $6$ to the power of $2$ .

$\sqrt{16 + 36}$

Add $16$ and $36$ .

$\sqrt{52}$

Rewrite $52$ as $2^{2} \cdot 13$ .

Factor $4$ out of $52$ .

$\sqrt{4 (13)}$

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

$\sqrt{2^{2} \cdot 13}$

Pull terms out from under the radical.

$2 \sqrt{13}$

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.

$(4 + 2 \sqrt{13}, 0)$

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.

$(4 - 2 \sqrt{13}, 0)$

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

$(4 + 2 \sqrt{13}, 0), (4 - 2 \sqrt{13}, 0)$

Step 6

Find the Foci 9x^2-4y^2-72x=0

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