Questions: Find an equation of the hyperbola that satisfies the given conditions. Vertices (0,0) and (0,-10), foci (0,1) and (0,-11)

Solution Steps

To find the equation of the hyperbola given the vertices and foci, we can follow these steps:

1. Identify the center of the hyperbola, which is the midpoint of the vertices.
2. Determine the distance between the center and the vertices (this is the value of 'a').
3. Determine the distance between the center and the foci (this is the value of 'c').
4. Use the relationship \(c^2 = a^2 + b^2\) to find 'b'.
5. Write the standard form of the hyperbola equation using the values of 'a' and 'b'.

The center of the hyperbola is the midpoint of the vertices \((0, 0)\) and \((0, -10)\). Thus, the center is calculated as: \[ \text{Center} = \left( \frac{0 + 0}{2}, \frac{0 + (-10)}{2} \right) = (0.0, -5.0) \]

The distance \(a\) from the center to a vertex is given by: \[ a = |y_{\text{vertex}} - y_{\text{center}}| = |0 - (-5)| = 5.0 \]

The distance \(c\) from the center to a focus is given by: \[ c = |y_{\text{focus}} - y_{\text{center}}| = |1 - (-5)| = 6.0 \]

Using the relationship \(c^2 = a^2 + b^2\), we can find \(b\): \[ c^2 = a^2 + b^2 \implies 6.0^2 = 5.0^2 + b^2 \implies 36 = 25 + b^2 \implies b^2 = 11 \implies b \approx 3.3166 \]

Since the hyperbola is vertical, the standard form of the equation is: \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \] Substituting \(h = 0\), \(k = -5\), \(a^2 = 25\), and \(b^2 \approx 11\): \[ \frac{(y + 5)^2}{25} - \frac{x^2}{11} = 1 \]

Final Answer