Hyperbola problems?
Question
About focus directrix and other information
Answer
The hyperbola is the least known and used of the conic sections. We seldom see a hyperbola in daily life, and it seldom appears in decoration or design. In spite of this, it has interesting properties and important applications. There is a literary term, hyperbole, that is the same word in Greek, meaning an excess. How the hyperbola acquired this name is related in Parabola, together with some general information on conic sections, and the focal definition of the hyperbola.
A hyperbola is sketched at the right. The origin is O, and the asymptotes form a symmetrical cross as shown. V and V’ are the vertices of the hyperbola, at a distance a on each side of the origin. Perpendicular lines from V and V’ define a rectangle by their points of intersection with the asymptotes, and the sides of this rectangle are a and b. Two parameters are required to specify a hyperbola, as for an ellipse. The slope of the asymptotes is |b/a|. Then, the hyperbola can be represented as the quadratic curve (x/a)2 – (y/b)2 = 1, the canonical equation of a hyperbola.
The foci F and F’ are located a distance c > a from the origin, where c is the hypotenuse of the right triangle whose sides are a and b. If you draw the reference rectangle for the hyperbola, the foci can be located quite simply by swinging an arc. The difference in the distances F’P and FP from the foci to any point P on the hyperbola is equal to 2a. It is not difficult to prove that this definition is equivalent to the canonical equation. Moreover, as the sketch indicates, the angle between FP and the normal to the hyperbola is equal to the angle between the normal and F’P, so a ray from F is reflected by the hyperbola so that it appears to be coming from the other focus. This is the analogue to the reflecting properties of the parabola and ellipse. The ratio c/a is the eccentricity of the hyperbola, and is > 1. We see that b = a(e2 – 1)1/2, and that the semi-latus rectum p = b2/a. The latter is derived from the right triangle with legs p and 2c, whose hypotenuse must be of length p + 2a from the focal definition.
