Explain why it is not possible for a hyperbola to have foci at (0,-2) and (0,2) and vertices at (0,-3) and (0,3).

Verified step by step guidance

1. <Hyperbolas have two foci and two vertices, and they are symmetric with respect to their center.>
2. <The center of a hyperbola is the midpoint of the line segment joining the vertices.>
3. <For the given vertices (0,-3) and (0,3), the center is at (0,0).>
4. <The distance from the center to each vertex is called the 'a' value, which is 3 in this case.>
5. <The distance from the center to each focus is called the 'c' value, which is 2 in this case. For a hyperbola, c must be greater than a, but here c < a, which is not possible.>

Key Concepts

Definition of a Hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center, and 'a' and 'b' are distances related to the vertices and foci.

Foci and Vertices of a Hyperbola

In a hyperbola, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The distance from the center to each vertex is 'a', while the distance from the center to each focus is 'c'. The relationship between these distances is given by the equation c² = a² + b², which must hold true for the hyperbola to be defined correctly.

Transverse Axis Orientation

The orientation of the transverse axis determines the shape and position of the hyperbola. For a hyperbola with a vertical transverse axis, the foci and vertices must have the same x-coordinate, while their y-coordinates differ. In the given question, the foci at (0,-2) and (0,2) suggest a vertical orientation, but the vertices at (0,-3) and (0,3) imply an inconsistency in the required distances, violating the properties of hyperbolas.