The correct hyperbola equation with center (0,0), focus (0,4), and vertex (0, -2) is y² - x² = 4, which is option (a).
The student has provided the center, focus, and vertex of a hyperbola and asks which equation represents this hyperbola. The correct equation must account for the center being at the origin (0,0), the focus along the y-axis at (0,4), and the vertex on the y-axis at (0, -2). Considering the information, we know that the hyperbola is vertical (opens up and down) since the focus and vertex values are on the y-axis.
Therefore, the correct form of the hyperbola's equation is y² - x² = 4, represented by option (a), which corresponds to the standard form of a hyperbola equation Ay² - Bx² = C, where A and B are positive and the hyperbola opens along the y-axis.