A hyperbola centered at the origin has vertices at (0,+-sqrt12) and foci at (0,+-sqrt45). Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at  (0,+-sqrt12) and foci at  (0,+-sqrt45). 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 centered at the origin with a vertical transverse axis. Standard form of equation for a hyperbola with vertical transverse axis:  (y-k)^2/a^2 - (x-h)^2/b^2 = 1 where (h, k) is the center of the hyperbola.Determining the Center: Determine the center (h, k) of the hyperbola. Since the hyperbola is centered at the origin, we have 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, ±√12), so the distance from the center to a vertex is √12. Therefore, a = √12.Finding the Semi-Minor Axis: Find the value of the semi-minor axis b using the relationship c^2 = a^2 + b^2, where c is the distance from the center to a focus. The foci are given at (0, ±√45), so c = √45. We already know that a = √12. We can now solve for b. c^2 = a^2 + b^2 (√45)^2 = (√12)^2 + b^2 45 = 12 + b^2 b^2 = 45 - 12 b^2 = 33 b = √33Writing 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 h = 0, k = 0, a = √12, and b = √33 into the standard form equation (y-k)^2/a^2 - (x-h)^2/b^2 = 1. (y - 0)^2/(√12)^2 - (x - 0)^2/(√33)^2 = 1 y^2/12 - x^2/33 = 1

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{12})$ and foci at $(0, \pm \sqrt{45})$ . $\newline$ Write the equation of this hyperbola.

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A hyperbola centered at the origin has vertices at (0,±12) (0, \pm \sqrt{12}) (0,±12​) and foci at (0,±45) (0, \pm \sqrt{45}) (0,±45​).\newlineWrite the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{12})$ and foci at $(0, \pm \sqrt{45})$ . $\newline$ Write the equation of this hyperbola.