A hyperbola centered at the origin has vertices at (0,+-sqrt6) and foci at (0,+-sqrt74). Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at  (0,+-sqrt6) and foci at  (0,+-sqrt74). Write the equation of this hyperbola.

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Standard Form of Hyperbola Equation: Identify the standard form of the equation for a hyperbola with a vertical transverse axis. Standard form of equation for a hyperbola with a vertical transverse axis:  (y-k)^2/a^2 - (x-h)^2/b^2 = 1Determining 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 at (0, ±√6), so a = √6.Calculating the Distance to the Foci: Calculate the distance c between the center and the foci. The foci are at (0, ±√74), so c = √74.Using the Relationship between a, b, and c: Use the relationship c^2 = a^2 + b^2 to find the value of b. Substitute the known values of a and c into the equation. c^2 = a^2 + b^2 √74^2 = √6^2 + b^2 74 = 6 + b^2 b^2 = 74 - 6 b^2 = 68 b = √68Writing the Equation in Standard Form: Write the equation of the hyperbola in standard form after substituting the values of h, k, a, and b. Substitute values of h, k, a, and b into (y-k)^2/a^2 - (x-h)^2/b^2 = 1. (y - 0)^2/(√6)^2 - (x - 0)^2/(√68)^2 = 1 y^2/6 - x^2/68 = 1

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{6})$ and foci at $(0, \pm \sqrt{74})$ . $\newline$ Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at (0,±6) (0, \pm \sqrt{6}) (0,±6​) and foci at (0,±74) (0, \pm \sqrt{74}) (0,±74​).\newlineWrite the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{6})$ and foci at $(0, \pm \sqrt{74})$ . $\newline$ Write the equation of this hyperbola.