Hyperbola

1. Definition of a Hyperbola
   A hyperbola is the set of all points on a plane where the absolute difference of the distances from two fixed points, and , is constant. These two points are called the foci of the hyperbola. If the line segment joining the two foci intersects the hyperbola at two points, labeled and , these points are called the vertices of the hyperbola.
   
   
   
   
   The line segment is called the transverse axis.
   The midpoint of is called the center of the hyperbola.
   
   
2. Equation of a Hyperbola
1. Horizontal Hyperbola
   For a hyperbola with foci at and , where the absolute difference of the distances from the foci to any point on the hyperbola is , the equation is:
   
   
2. Vertical Hyperbola
   For a hyperbola with foci at and , where the absolute difference of the distances from the foci to any point on the hyperbola is , the equation is:
   
   
3. Asymptotes of a Hyperbola
   For the hyperbola :
1. The equations of the asymptotes are:
2. The two asymptotes are perpendicular to each other when










4. Translation of a Hyperbola
   If the hyperbola is translated by units horizontally and units vertically, the new equation becomes:
5. Asymptotes of a Translated Hyperbola
   The equations of the asymptotes for the translated hyperbola are:

Conic Sections

A conic section is a curve represented by a second-degree equation in and that cannot be factored into the product of two linear equations. The general equation is: (where and are constants.)

Circles, parabolas, ellipses, and hyperbolas are all examples of conic sections.