Question

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).

Accepted Answer

A hyperbola is defined as the set of all points where the absolute difference of the distances to two fixed points (the foci) is constant. The vertices of the hyperbola are points on the hyperbola that lie along the transverse axis, which is the axis that passes through the foci. For a hyperbola centered at the origin with a vertical transverse axis, the standard equation is: y2a2 - x2b2 = 1, where a is the distance from the center to each vertex, and c is the distance from the center to each focus. The relationship between these values is given by $$c^2$$ = $$a^2$$ + $$b^2. In $$this problem, the foci are at (0, -2) and (0, 2), so the distance from the center (0, 0) to each focus is c = 2. The vertices are at (0, -3) and (0, 3), so the distance from the center to each vertex is a = 3. Using the relationship $$c^2$$ = $$a^2$$ + $$b^2$$, substitute c = 2 and a = 3: $$2^2$$ = $$3^2$$ + $$b^2. $$Simplify this equation to find $$b^2. $$After simplifying, you will find that $$b^2$$ becomes negative, which is not possible because $$b^2$$ represents the square of a real number. This contradiction indicates that the given configuration of foci and vertices is not possible for a hyperbola.

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 video answer for a similar problem:

Key Concepts

Definition of a Hyperbola

Foci and Vertices Relationship

Geometric Configuration of Foci and Vertices