What is the shape of a hyperbola?

The shape of a hyperbola consists of two distinct branches that open away from each other. These branches are located in opposite quadrants of the Cartesian plane, depending on the orientation of the hyperbola. For example, a hyperbola that is centered at the origin with a horizontal transverse axis will typically have one branch in the first quadrant and the other in the third quadrant. Conversely, if it has a vertical transverse axis, one branch will be in the second quadrant and the other in the fourth quadrant.

In contrast to closed shapes like ellipses or circles, a hyperbola is characterized by its open shape, which means that the branches never intersect and extend infinitely in their respective directions. Thus, the correct answer highlights this defining feature of hyperbolas, affirming that they consist of two separate branches located in opposite quadrants instead of existing as a singular or closed curve.