__PROPERTIES OF A HYPERBOLA__

**In
this section we will deal with some properties which are unique to the
hyperbola. Shown here are some terms used when dealing with hyperbolas.
Asymptotes are the lines which, as they extend to infinity with the curve,
approach nearer and nearer the curve but never actually touch it. The line
joining the vertices is called the transverse axis and the line perpendicular to
this is called the conjugate axis.**

**PROPERTY 1**

**The
asymptotes of a hyperbola lie on the points of intersection of circle containing
the foci and tangents from the vertices.**

**PROPERTY 2**

**The
directrix lies on the point of intersection of the auxiliary circle and the
asymptotes**

**PROPERTY 3**

**If
P is any point on the curve and PR and PS are drawn parallel to the asymptotes,**

**THEN PR
x PS = A CONSTANT**

**This
important property is characteristic of the hyperbola. There is a method for
constructing a hyperbola shown below where the asymptotes and a point p on the
curve are given and it uses this property to find the curve.**