Definition and properties of a hyperbola

A hyperbola is a type of conic section that is defined as the set of all points (x, y) in a plane where the difference of the distances between two fixed points called foci is constant. This constant difference is equal to the distance between the vertices of the hyperbola.

Properties of a Hyperbola:

1. Symmetry: A hyperbola is symmetric with respect to both its axes. This means if you were to fold the graph along either axis, the two halves would match up perfectly.

2. Asymptotes: A hyperbola has two asymptotes. These are lines that the hyperbola approaches but never touches. The asymptotes intersect at the center of the hyperbola and form a right angle.

3. Foci: Each hyperbola has two foci. The foci always lie on the major axis, and they are equidistant from the center of the hyperbola.

4. Vertices: The vertices of a hyperbola are the points where it intersects its major axis. The distance between the vertices is less than the distance between the foci.

5. Eccentricity: The eccentricity of a hyperbola is always greater than 1. It is a measure of how "stretched out" the hyperbola is. The greater the eccentricity, the more "stretched out" the hyperbola appears.

6. Conjugate Axis: The conjugate axis of a hyperbola is the line segment perpendicular to the transverse axis through the center of the hyperbola. The ends of the conjugate axis are called the co-vertices of the hyperbola.

7. Transverse Axis: The transverse axis of a hyperbola is the line segment that contains the vertices and foci of the hyperbola. The length of the transverse axis is equal to the distance between the vertices of the hyperbola.