## Conic Sections: Hyperbola

A hyperbola is defined as the locus of points where the difference in the distance to two fixed points (called foci) is constant. That fixed difference in distance is two times

*a*where

*a*is the distance from the center of the hyperbola to the vertex of the nearest branch of the hyperbola.

*a*is also known as the semi-major axis of the hyperbola. The foci lie on the transverse axis and their midpoint is called the center.

In an x-y coordinate system, the hyperbola opening east-west with center (h,k) along with semi-major axis, a, and semi-minor axis b is represented by:

In
an x-y
coordinate system, the hyperbola opening north-south with center (h,k) along with
semi-major axis, a,
and semi-minor axis b
is represented by:

In the above figure, the blue colored hyperbola opens east-west while the light green colored hyperbola opens north-south.

Eccentricity of Hyperbola:

The foci for an east-west opening hyperbola are given by

(h ± c, k) where *c*^{2}
= *a*^{2} + *b*^{2}

and for a north-south opening hyperbola are given by

(h,
k ± c) where
*c*^{2}
= *a*^{2} + *b*^{2}

### Hyperbolas In Polar Coordinates:

*East-west
opening hyperbola:*

r^{2}
= a sec 2θ

*North-south
opening hyperbola:*

r^{2}
= −a sec 2θ

*Northeast-southwest
opening hyperbola:*

r^{2}
= a csc 2θ

*Northwest-southeast
opening hyperbola:*

r^{2}
= −a csc 2θ

In all formulas the center is at the pole, and *a*
is the semi-major axis and semi-minor axis.

### Hyperbolas In Parametric Form:

*East-west
opening hyperbola:*

*x
= a *sec* t + h, y = b *tan
*t + k*

or

*x
= ± a *cosh* t + h, y = b *sinh
*t + k*

*North-south
opening hyperbola:*

*x
= a *tan* t + h, y = b *sec
*t + k*

or

*x
= a *sinh* t + h, y = **±**b *cosh
*t + k*

In all formulae (*h*,*k*)
is
the center of the hyperbola, *a* is the semi-major
axis, and *b* is the semi-minor axis.