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A hyperbola centered at the origin has vertices at
(0,+-sqrt54) and foci at
(0,+-sqrt89).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{54})$ and foci at $(0, \pm \sqrt{89})$ . $\newline$ 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{54})

and foci at

(0, \pm \sqrt{89})

.

\newline

Write the equation of this hyperbola.

Standard form of hyperbola equation: 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: $\newline$ $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$
Determining the center: Determine the center $(h, k)$ of the hyperbola. Since the hyperbola is centered at the origin, $h = 0$ and $k = 0$ .
Finding the semi-major axis: Find the value of the semi-major axis $a$ . The vertices are given at $(0, \pm\sqrt{54})$ , so $a = \sqrt{54}$ .
Calculating $a^2$ : Calculate the value of $a^2$ . $a^2 = (\sqrt{54})^2 = 54$ .
Finding the 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 to the foci $c$ for a hyperbola is $c^2 = a^2 + b^2$ .
Calculating $c$ : Calculate the value of $c$ using the coordinates of the foci. The foci are given at $(0, \pm\sqrt{89})$ , so $c = \sqrt{89}$ .
Calculating $c^2$ : Calculate the value of $c^2$ . $c^2 = (\sqrt{89})^2 = 89$ .
Using $c^2$ to find $b^2$ : Use the relationship $c^2 = a^2 + b^2$ to find $b^2$ . Substitute the known values of $a^2$ and $c^2$ into the equation. $89 = 54 + b^2$ $b^2 = 89 - 54$ $b^2 = 35$
Writing the equation in standard form: Write the equation of the hyperbola in standard form after substituting the values of $h$ , $k$ , $a^2$ , and $b^2$ . Substitute $h = 0$ , $k = 0$ , $a^2 = 54$ , and $b^2 = 35$ into $(y-k)^2/a^2 - (x-h)^2/b^2 = 1$ . $(y - 0)^2/54 - (x - 0)^2/35 = 1$ $k$ $0$

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