A hyperbola centered at the origin has vertices at (+-sqrt45,0) and foci at (+-sqrt70,0). 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. Standard form of equation for a hyperbola with horizontal transverse axis: (x^2/a^2) - (y^2/b^2) = 1Determine a and a^2: Determine the values of a and a^2. The vertices are at (±√45,0), so a = √45 and a^2 = 45.Determine c and c^2: Determine the values of c and c^2. The foci are at (±√70,0), so c = √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 = 25Write 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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Algebra 2

Write equations of hyperbolas in standard form using properties

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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 $(x^2/a^2) - (y^2/b^2) = 1$ . $(x^2/45) - (y^2/25) = 1$