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Multiple Choice
Find the equation for a hyperbola with a center at , focus at and vertex at .
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Identify the orientation of the hyperbola. Since the center is at (0,0) and the focus is at (0,-6), the hyperbola is vertical, meaning the equation will be of the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
Determine the distance from the center to the vertex, which is the value of \( a \). The vertex is at (0,4), so \( a = 4 \). Therefore, \( a^2 = 16 \).
Determine the distance from the center to the focus, which is the value of \( c \). The focus is at (0,-6), so \( c = 6 \).
Use the relationship \( c^2 = a^2 + b^2 \) to find \( b^2 \). Substitute \( c = 6 \) and \( a^2 = 16 \) into the equation: \( 36 = 16 + b^2 \). Solve for \( b^2 \).
Substitute \( a^2 = 16 \) and the calculated \( b^2 \) into the hyperbola equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) to get the final equation of the hyperbola.