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.
The asymptotes of a hyperbola lie on the points of intersection of circle containing the foci and tangents from the vertices.
The directrix lies on the point of intersection of the auxiliary circle and the asymptotes
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.