Final answer:
The equation representing a hyperbola with a center at (0, 0), a vertex at (−48, 0), and a focus at (50, 0) is x²/48² - y²/14² = 1.
Step-by-step explanation:
To determine which equation represents a hyperbola with the given features, we can use the standard form of a hyperbola's equation. For a hyperbola centered at the origin (0,0) with its transverse axis along the x-axis, the equation is x²/a² - y²/b² = 1, where a is the distance from the center to a vertex and b is related to the distance from the center to a focus through the relationship c² = a² + b² where c is the distance from the center to a focus.
If a vertex is at (-48, 0), then a = 48. For a focus at (50, 0), we have c = 50. Plugging these values into c² = a² + b² gives us 50² = 48² + b², which can be solved to find b². Thus, b² = 50² - 48² = 2500 - 2304 = 196, which means b = 14. The corresponding hyperbola equation is x²/48² - y²/14² = 1.