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A hyperbola centered at the origin has vertices at
(+-sqrt7,0) and foci at
(+-sqrt27,0).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $( \pm \sqrt{7}, 0)$ and foci at $( \pm \sqrt{27}, 0)$ . $\newline$ Write the equation of this hyperbola.

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Convert equations of conic sections from general to standard form

Full solution

Q. A hyperbola centered at the origin has vertices at

( \pm \sqrt{7}, 0)

and foci at

( \pm \sqrt{27}, 0)

.

\newline

Write the equation of this hyperbola.

Hyperbola Equation: The standard form of the equation of a hyperbola centered at the origin with horizontal transverse axis is $(x^2/a^2) - (y^2/b^2) = 1$ , where $2a$ is the distance between the vertices and $2c$ is the distance between the foci. We are given the vertices at $(\pm\sqrt{7}, 0)$ , which means $a = \sqrt{7}$ . We are also given the foci at $(\pm\sqrt{27}, 0)$ , which means $c = \sqrt{27}$ .
Finding Value of b: We need to find the value of $b$ . The relationship between $a$ , $b$ , and $c$ in a hyperbola is $c^2 = a^2 + b^2$ . We already know that $a = \sqrt{7}$ and $c = \sqrt{27}$ . Let's plug these values into the equation to find $b^2$ . $\newline$ $c^2 = a^2 + b^2$ $\newline$ $(\sqrt{27})^2 = (\sqrt{7})^2 + b^2$ $\newline$ $a$ $0$ $\newline$ $a$ $1$ $\newline$ $a$ $2$
Writing Hyperbola Equation: Now that we have $b^2 = 20$ , we can write the equation of the hyperbola. The equation is $(\frac{x^2}{a^2}) - (\frac{y^2}{b^2}) = 1$ . Substituting $a^2 = 7$ and $b^2 = 20$ into the equation, we get: $(\frac{x^2}{7}) - (\frac{y^2}{20}) = 1$

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