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A hyperbola centered at the origin has vertices at
(+-sqrt4,0) and foci at
(+-sqrt9,0).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $( \pm \sqrt{4}, 0)$ and foci at $( \pm \sqrt{9}, 0)$ . $\newline$ 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

( \pm \sqrt{4}, 0)

and foci at

( \pm \sqrt{9}, 0)

.

\newline

Write the equation of this hyperbola.

Identify standard form: Identify the standard form of the equation for a hyperbola centered at the origin with horizontal transverse axis. $\newline$ Standard form of equation for a hyperbola with horizontal transverse axis: $\newline$ $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$ where $(h, k)$ is the center of the hyperbola.
Determine center coordinates: Determine the values of $h$ and $k$ for the center of the hyperbola. Since the hyperbola is centered at the origin, $h = 0$ and $k = 0$ .
Find semi-major axis: Find the value of the semi-major axis $a$ . The vertices are given at $(\pm\sqrt{4},0)$ , which means $a = \sqrt{4} = 2$ .
Find 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 $(\pm\sqrt{9},0)$ , which means $c = \sqrt{9} = 3$ .
Calculate value of b: Calculate the value of $b$ using the relationship $c^2 = a^2 + b^2$ . Substitute $a = 2$ and $c = 3$ into the equation to find $b$ . $3^2 = 2^2 + b^2$ $9 = 4 + b^2$ $b^2 = 9 - 4$ $b^2 = 5$ $b = \sqrt{5}$
Write equation in standard form: Write the equation of the hyperbola in standard form after substituting the values of $h$ , $k$ , $a$ , and $b$ . Substitute $h = 0$ , $k = 0$ , $a = 2$ , and $b = \sqrt{5}$ into $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$ . $x^2/2^2 - y^2/(\sqrt{5})^2 = 1$ $k$ $0$

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