Question: Determine the equation of the hyperbola with vertices (0, -4) and (0, 4) and foci (0, -6) and (0,…

Determine the equation of the hyperbola with vertices (0, -4) and (0, 4) and foci (0, -6) and (0, 6).

Solution

To find the equation of the hyperbola, follow these steps: 1. Identify the center of the hyperbola. The center is the midpoint of the line segment joining the vertices. The vertices are \((0, -4)\) and \((0, 4)\). \[ \text{Midpoint} = \left( \frac{0+0}{2}, \frac{-4+4}{2} \right) = (0, 0) \] 2. Determine the distance between the vertices to find \(2a\). The vertex distance is the length of the segment joining the vertices. \[ 2a = 4 - (-4) = 8 \quad \Rightarrow \quad a = 4 \] 3. Calculate the distance between the foci to find \(2c\). The foci are at \((0, -6)\) and \((0, 6)\). \[ 2c = 6 - (-6) = 12 \quad \Rightarrow \quad c = 6 \] 4. Use the relationship \(c^2 = a^2 + b^2\) to find \(b^2\). \[ c^2 = 6^2 = 36 \] \[ a^2 = 4^2 = 16 \] \[ b^2 = c^2 - a^2 = 36 - 16 = 20 \] 5. Write the equation of the hyperbola. Since the transverse axis is vertical, the equation is: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] Substituting the known values for \(a^2\) and \(b^2\): \[ \frac{y^2}{16} - \frac{x^2}{20} = 1 \] This is the equation of the hyperbola.