Hyperbola

The hyperbola is the locus of points on the plane whose difference of distances to two fixed points, foci, are constant.

Elements of the Hyperbola

Foci

The foci are the fixed points of the hyperbola. They are denoted by F and F'.

Transverse Axis or real axis

The tranverse axis is the line segment between the foci.

Conjugate Axis or imaginary axis

The conjuagate axis is the perpendicular bisector of the line segment (transverse axis).

Center

The center is the point of intersection of the axes and is also the center of symmetry of the hyperbola.

Vertices

The points A and A' are the points of intersection of the hyperbola with the transverse axis.

The focal radii are the line segments that join a point on the hyperbola with the foci: PF and PF'.

Focal Length

The focal length is the line segment , which has a length of 2c.

Semi-Major Axis

The semi-major axis is the line segment that runs from the center to a vertex of the hyperbola. Its length is a.

Semi-Minor Axis

The semi-minor axis is a line segment which is perpendicular to the semi-major axis. Its length is b.

Axes of Symmetry

The axes of symmetry are the lines that coincide with the transversal and conjugate axis.

Asymptotes

The asymptotes are the lines with the equations:

The Relationship Between the Semiaxes:

Eccentricity of a Hyperbola:

Equation of the Hyperbola

Hyperbolas Centered at (0, 0)

Horizontal Transverse Axis

F'(−c, 0) and F(c, 0)

Vertical Transverse Axis

F'(0, −c) and F(0, c)

Hyperbolas Centered at (x0, y0)

Horizontal Transverse Axis

F(x0+c, y0) and F'(x0− c, y0)

By removing the denominators, an equation is obtained in the form:

Keep in mind that A and B have opposite signs.

Vertical Transverse Axis

F(x0, y0+c) and F'(x0, y0−c)

Rectangular Hyperbola

Hyperbolas where the semiaxes are equal are called rectangular or equilateral hyperbolas (a = b).

The equation of a rectangular hyperbola is:

The equations of the asymptotes are:

,

That is, the angle bisectors of the quadrants.

The eccentricity is:

Equation of a Rectangular Hyperbola

To switch the asymptotes to those determined by the x and y-axis, turn the asymptote −45° about the origin.

If it is rotated 45°, the hyperbola is in the second and fourth quadrant.