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A hyperbola centered at the origin has vertices at
(0,+-sqrt12) and foci at
(0,+-sqrt45).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{12})$ and foci at $(0, \pm \sqrt{45})$ . $\newline$ Write the equation of this hyperbola.

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Write equations of hyperbolas in standard form using properties

Full solution

Q. A hyperbola centered at the origin has vertices at

(0, \pm \sqrt{12})

and foci at

(0, \pm \sqrt{45})

.

\newline

Write the equation of this hyperbola.

Standard Form of Hyperbola Equation: Identify the standard form of the equation for a hyperbola centered at the origin with a vertical transverse axis. $\newline$ Standard form of equation for a hyperbola with vertical transverse axis: $\newline$ $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$ where $(h, k)$ is the center of the hyperbola.
Determining the Center: Determine the center $(h, k)$ of the hyperbola. Since the hyperbola is centered at the origin, we have $h = 0$ and $k = 0$ .
Finding the Semi-Major Axis: Find the value of the semi-major axis $a$ . The vertices are given at $(0, \pm\sqrt{12})$ , so the distance from the center to a vertex is $\sqrt{12}$ . Therefore, $a = \sqrt{12}$ .
Finding the Semi-Minor Axis: Find the value of the semi-minor axis $b$ using the relationship $c^2 = a^2 + b^2$ , where $c$ is the distance from the center to a focus. $\newline$ The foci are given at $(0, \pm\sqrt{45})$ , so $c = \sqrt{45}$ . We already know that $a = \sqrt{12}$ . We can now solve for $b$ . $\newline$ $c^2 = a^2 + b^2$ $\newline$ $(\sqrt{45})^2 = (\sqrt{12})^2 + b^2$ $\newline$ $45 = 12 + b^2$ $\newline$ $c^2 = a^2 + b^2$ $0$ $\newline$ $c^2 = a^2 + b^2$ $1$ $\newline$ $c^2 = a^2 + b^2$ $2$
Writing the Equation in Standard Form: Write the equation of the hyperbola in standard form after substituting the values of $h$ , $k$ , $a$ , and $b$ . Substitute $h = 0$ , $k = 0$ , $a = \sqrt{12}$ , and $b = \sqrt{33}$ into the standard form equation $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$ . $(y - 0)^2/(\sqrt{12})^2 - (x - 0)^2/(\sqrt{33})^2 = 1$ $k$ $0$

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