Given the foci of a hyperbola are at (0, 5) and (0, -5), and the distance from the center to one vertex is 4, find the equation of the hyperbola.
Answer
Step 5: Therefore, the equation of the hyperbola is \(\frac{(y-0)^2}{4^2} - \frac{(x-0)^2}{3^2} = 1\), or simplified to \(\frac{y^2}{16} - \frac{x^2}{9} = 1\).
Step 1 ：Step 1: The distance between the two foci is 10, so \(2c = 10\) and \(c = 5\).
Step 2 ：Step 2: The distance from the center to one vertex is 4, so \(a = 4\).
Step 3 ：Step 3: Since the foci are vertical, the hyperbola is vertical. Hence, the equation of the hyperbola is of the form \(\frac{(y-h)^2}{a^2} - \frac{(x-k)^2}{b^2} = 1\).
Step 4 ：Step 4: We also know that \(c^2 = a^2 + b^2\). Substituting the values of \(a\) and \(c\) we get \(b^2 = c^2 - a^2 = 5^2 - 4^2 = 9\).
Step 5 ：Step 5: Therefore, the equation of the hyperbola is \(\frac{(y-0)^2}{4^2} - \frac{(x-0)^2}{3^2} = 1\), or simplified to \(\frac{y^2}{16} - \frac{x^2}{9} = 1\).
