A hyperbola centered at the origin has vertices at (+-sqrt61,0) and foci at (+-sqrt98,0). Write the equation of this hyperbola.

Identify standard 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 values of a and a^2: Determine the values of a and a^2. The vertices are at (±√61,0), so a = √61 and a^2 = 61.Determine values of c and c^2: Determine the values of c and c^2. The foci are at (±√98,0), so c = √98 and c^2 = 98.Use relationship c^2 = a^2 + 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. 98 = 61 + b^2 b^2 = 98 - 61 b^2 = 37Write equation in standard form: Write the equation of the hyperbola in standard form using the values of a^2 and b^2. Substitute a^2 = 61 and b^2 = 37 into the standard form equation (x^2/a^2) - (y^2/b^2) = 1. (x^2/61) - (y^2/37) = 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{61}, 0)

and foci at

( \pm \sqrt{98}, 0)

\newline

Write the equation of this hyperbola.

Identify standard 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 values of $a$ and $a^2$ : Determine the values of $a$ and $a^2$ . The vertices are at $(\pm\sqrt{61},0)$ , so $a = \sqrt{61}$ and $a^2 = 61$ .
Determine values of $c$ and $c^2$ : Determine the values of $c$ and $c^2$ . The foci are at $(\pm\sqrt{98},0)$ , so $c = \sqrt{98}$ and $c^2 = 98$ .
Use relationship $c^2 = a^2 + b^2$ : 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. $98 = 61 + b^2$ $b^2 = 98 - 61$ $b^2 = 37$
Write equation in standard form: Write the equation of the hyperbola in standard form using the values of $a^2$ and $b^2$ . $\newline$ Substitute $a^2 = 61$ and $b^2 = 37$ into the standard form equation $(x^2/a^2) - (y^2/b^2) = 1$ . $\newline$ $(x^2/61) - (y^2/37) = 1$