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A hyperbola centered at the origin has vertices at
(0,+-sqrt19) and foci at
(0,+-sqrt55).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{19})$ and foci at $(0, \pm \sqrt{55})$ . $\newline$ 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{19})

and foci at

(0, \pm \sqrt{55})

.

\newline

Write the equation of this hyperbola.

Identify standard form: Identify the standard form of the equation for a hyperbola with a vertical transverse axis. $\newline$ Standard form of equation for a hyperbola with a vertical transverse axis: $\newline$ $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$
Determine center: Determine the values of $h$ and $k$ , which represent 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 $a$ . $a$ is the distance from the center to a vertex along the y-axis. The vertices are given as $(0, \pm\sqrt{19})$ , so $a = \sqrt{19}$ .
Find semi-minor axis: Find the value of the semi-minor axis $b$ . The relationship between the semi-major axis $a$ , the semi-minor axis $b$ , and the distance $c$ from the center to a focus is $c^2 = a^2 + b^2$ . The foci are given as $(0, \pm\sqrt{55})$ , so $c = \sqrt{55}$ .
Calculate value of b: Calculate the value of $b$ using the relationship $c^2 = a^2 + b^2$ .
$c^2 = (\sqrt{55})^2$
$c^2 = 55$
$a^2 = (\sqrt{19})^2$
$a^2 = 19$
$b^2 = c^2 - a^2$
$b^2 = 55 - 19$
$b^2 = 36$
$b = \sqrt{36}$
$c^2 = a^2 + b^2$ $0$
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 values of $h$ , $k$ , $a$ , and $b$ into $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$ . $(y - 0)^2/\sqrt{19}^2 - (x - 0)^2/6^2 = 1$ $k$ $0$

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