A hyperbola centered at the origin has vertices at (0,+-sqrt21) and foci at (0,+-sqrt34). Write the equation of this hyperbola.

Question

A hyperbola centered at the origin has vertices at  (0,+-sqrt21) and foci at  (0,+-sqrt34). Write the equation of this hyperbola.

Bytelearn · Accepted Answer

Identify standard form: 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.Determine center coordinates: Determine the values of h and k for the center of the hyperbola. Since the hyperbola is centered at the origin, h = 0 and k = 0.Find semi-major axis: Find the value of the semi-major axis a. The vertices are at (0, ±√21), so the distance from the center to a vertex is a = √21.Find 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. The foci are at (0, ±√34), so c = √34.Calculate b using c^2 = a^2 + b^2: Calculate the value of b using the relationship c^2 = a^2 + b^2. c^2 = (√34)^2 c^2 = 34 a^2 = (√21)^2 a^2 = 21 Substitute a^2 and c^2 into the relationship to find b^2. 34 = 21 + b^2 b^2 = 34 - 21 b^2 = 13Write 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^2 = 21, and b^2 = 13 into (y-k)^2/a^2 - (x-h)^2/b^2 = 1. (y - 0)^2/21 - (x - 0)^2/13 = 1 y^2/21 - x^2/13 = 1

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{21})$ and foci at $(0, \pm \sqrt{34})$ . $\newline$ Write the equation of this hyperbola.

Full solution

More problems from Write equations of hyperbolas in standard form using properties

Related topics

A hyperbola centered at the origin has vertices at (0,±21) (0, \pm \sqrt{21}) (0,±21​) and foci at (0,±34) (0, \pm \sqrt{34}) (0,±34​).\newlineWrite the equation of this hyperbola.

A hyperbola centered at the origin has vertices at $(0, \pm \sqrt{21})$ and foci at $(0, \pm \sqrt{34})$ . $\newline$ Write the equation of this hyperbola.