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A hyperbola centered at the origin has vertices at
(+-sqrt45,0) and foci at
(+-sqrt70,0).
Write the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $( \pm \sqrt{45}, 0)$ and foci at $( \pm \sqrt{70}, 0)$ . $\newline$ 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{45}, 0)

and foci at

( \pm \sqrt{70}, 0)

.

\newline

Write the equation of this hyperbola.

Identify Equation 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: $\newline$ $(\frac{x^2}{a^2}) - (\frac{y^2}{b^2}) = 1$
Determine $a$ and $a^2$ : Determine the values of $a$ and $a^2$ . The vertices are at $(\pm\sqrt{45},0)$ , so $a = \sqrt{45}$ and $a^2 = 45$ .
Determine $c$ and $c^2$ : Determine the values of $c$ and $c^2$ . The foci are at $(\pm\sqrt{70},0)$ , so $c = \sqrt{70}$ and $c^2 = 70$ .
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. $70 = 45 + b^2$ $b^2 = 70 - 45$ $b^2 = 25$
Write Standard Form Equation: Write the equation of the hyperbola in standard form using the values of $a^2$ and $b^2$ . Substitute $a^2 = 45$ and $b^2 = 25$ into the standard form equation $(x^2/a^2) - (y^2/b^2) = 1$ . $(x^2/45) - (y^2/25) = 1$

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