Find the Properties (y^2)/49-(x^2)/64=1

$\frac{y^{2}}{49} - \frac{x^{2}}{64} = 1$

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 $1$ .

$\frac{y^{2}}{49} - \frac{x^{2}}{64} = 1$

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

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

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, $a$ .

$a = 7$

$b = 8$

$k = 0$

$h = 0$

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

$(0, 0)$

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.

$\sqrt{{(7)}^{2} + {(8)}^{2}}$

Simplify.

Raise $7$ to the power of $2$ .

$\sqrt{49 + {(8)}^{2}}$

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

$\sqrt{49 + 64}$

Add $49$ and $64$ .

$\sqrt{113}$

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.

$(0, 7)$

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.

$(0, - 7)$

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

$(0, 7), (0, - 7)$

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.

$(0, \sqrt{113})$

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.

$(0, - \sqrt{113})$

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

$(0, \sqrt{113}), (0, - \sqrt{113})$

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.

$\frac{\sqrt{{(7)}^{2} + {(8)}^{2}}}{7}$

Simplify the numerator.

Raise $7$ to the power of $2$ .

$\frac{\sqrt{49 + 8^{2}}}{7}$

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

$\frac{\sqrt{49 + 64}}{7}$

Add $49$ and $64$ .

$\frac{\sqrt{113}}{7}$

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.

$\frac{8^{2}}{\sqrt{113}}$

Simplify.

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

$\frac{64}{\sqrt{113}}$

Multiply $\frac{64}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\frac{64}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}}$

Combine and simplify the denominator.

Multiply $\frac{64}{\sqrt{113}}$ by $\frac{\sqrt{113}}{\sqrt{113}}$ .

$\frac{64 \sqrt{113}}{\sqrt{113} \sqrt{113}}$

Raise $\sqrt{113}$ to the power of $1$ .

$\frac{64 \sqrt{113}}{{\sqrt{113}}^{1} \sqrt{113}}$

Raise $\sqrt{113}$ to the power of $1$ .

$\frac{64 \sqrt{113}}{{\sqrt{113}}^{1} {\sqrt{113}}^{1}}$

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

$\frac{64 \sqrt{113}}{{\sqrt{113}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{64 \sqrt{113}}{{\sqrt{113}}^{2}}$

Rewrite ${\sqrt{113}}^{2}$ as $113$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{113}$ as $113^{\frac{1}{2}}$ .

$\frac{64 \sqrt{113}}{{(113^{\frac{1}{2}})}^{2}}$

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

$\frac{64 \sqrt{113}}{113^{\frac{1}{2} \cdot 2}}$

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

$\frac{64 \sqrt{113}}{113^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{64 \sqrt{113}}{113^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{64 \sqrt{113}}{113^{1}}$

Evaluate the exponent.

$\frac{64 \sqrt{113}}{113}$

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

$y = \pm \frac{7}{8} x + 0$

Simplify $\frac{7}{8} x + 0$ .

Add $\frac{7}{8} x$ and $0$ .

$y = \frac{7}{8} x$

Combine $\frac{7}{8}$ and $x$ .

$y = \frac{7 x}{8}$

Simplify $- \frac{7}{8} x + 0$ .

Add $- \frac{7}{8} x$ and $0$ .

$y = - \frac{7}{8} x$

Combine $x$ and $\frac{7}{8}$ .

$y = - \frac{x \cdot 7}{8}$

Move $7$ to the left of $x$ .

$y = - \frac{7 x}{8}$

This hyperbola has two asymptotes.

$y = \frac{7 x}{8}, y = - \frac{7 x}{8}$

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

Center: $(0, 0)$

Vertices: $(0, 7), (0, - 7)$

Foci: $(0, \sqrt{113}), (0, - \sqrt{113})$

Eccentricity: $\frac{\sqrt{113}}{7}$

Focal Parameter: $\frac{64 \sqrt{113}}{113}$

Asymptotes: $y = \frac{7 x}{8}$ , $y = - \frac{7 x}{8}$

Find the Properties (y^2)/49-(x^2)/64=1

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