A hyperbola centered at the origin has vertices at (0,+-sqrt54) and foci at (0,+-sqrt89). Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at  (0,+-sqrt54) and foci at  (0,+-sqrt89). 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 given at (0, ±√54), so a = √54.Calculating a^2: Calculate the value of a^2. a^2 = (√54)^2 = 54.Finding the semi-minor axis: Find the value of the semi-minor axis b. The relationship between the semi-major axis a, the semi-minor axis b, and the distance to the foci c for a hyperbola is c^2 = a^2 + b^2.Calculating c: Calculate the value of c using the coordinates of the foci. The foci are given at (0, ±√89), so c = √89.Calculating c^2: Calculate the value of c^2. c^2 = (√89)^2 = 89.Using c^2 to 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. 89 = 54 + b^2 b^2 = 89 - 54 b^2 = 35Writing the equation in standard form: Write the equation of the hyperbola in standard form after substituting the values of h, k, a^2, and b^2. Substitute h = 0, k = 0, a^2 = 54, and b^2 = 35 into (y-k)^2/a^2 - (x-h)^2/b^2 = 1. (y - 0)^2/54 - (x - 0)^2/35 = 1 y^2/54 - x^2/35 = 1

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{54})$ and foci at $(0, \pm \sqrt{89})$ . $\newline$ Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at (0,±54) (0, \pm \sqrt{54}) (0,±54​) and foci at (0,±89) (0, \pm \sqrt{89}) (0,±89​).\newlineWrite the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{54})$ and foci at $(0, \pm \sqrt{89})$ . $\newline$ Write the equation of this hyperbola.