Final answer:
The correct equation for the hyperbola with the given conditions is not presented in the student's options. We calculate b² = 196 from the vertex and focus information, but none of the provided options match the correct equation x²/2304 - y²/196 = 1.
Step-by-step explanation:
The equation that represents a hyperbola with a center at (0, 0), a vertex at (-48, 0), and a focus at (50, 0) can be found by examining the standard form of a hyperbola and applying the given information. For hyperbolas centered at the origin (0,0), the standard forms are:
- Horizontal hyperbola: (x²/a²) - (y²/b²) = 1
- Vertical hyperbola: (y²/a²) - (x²/b²) = 1
Here, the hyperbola is horizontal since the vertex and focus lie along the x-axis.
The distance from the center to a vertex, a, is 48, so a² = 48² = 2304. The distance from the center to a focus, c, is 50, so we have c² = 50² = 2500. The relationship between a, b, and c in a hyperbola is c² = a² + b², so b² = c² - a² = 2500 - 2304 = 196. The correct equation for the hyperbola is therefore (x²/2304) - (y²/196) = 1 which simplifies to x²/2304 - y²/196 = 1, giving us x²/2400 - y²/2400 = 1 which is not presented in any of the options provided. Thus, a correct hyperbola equation with the given specifications is not listed among the choices.