The hyperbola's _____ is the point halfway between its two vertices.A.centerB.focusC.directrix

The correct answer is A. center.

The center of a hyperbola is indeed the point that lies exactly halfway between its two vertices. To understand this better, let’s visualize the hyperbola, which consists of two separate branches opening either horizontally or vertically. Each branch has two vertices, which are the points where the hyperbola intersects its transverse axis. The center is a crucial point in the geometry of the hyperbola as it serves as the origin for the hyperbola's standard form equations.

In more technical terms, the center is located at the midpoint of the line segment connecting the two vertices. Therefore, it is a vital point for defining the characteristics and properties of the hyperbola, including its asymptotes and the overall shape of the graph. Meanwhile, focus points and directrices are also significant features of a hyperbola, but they serve different roles in the hyperbola's structure.