7.5 Conic sections  (Page 7/23)

Hyperbolas also have interesting reflective properties. A ray directed toward one focus of a hyperbola is reflected by a hyperbolic mirror toward the other focus. This concept is illustrated in the following figure.

This property of the hyperbola has important applications. It is used in radio direction finding (since the difference in signals from two towers is constant along hyperbolas), and in the construction of mirrors inside telescopes (to reflect light coming from the parabolic mirror to the eyepiece). Another interesting fact about hyperbolas is that for a comet entering the solar system, if the speed is great enough to escape the Sun’s gravitational pull, then the path that the comet takes as it passes through the solar system is hyperbolic.

Eccentricity and directrix

An alternative way to describe a conic section involves the directrices, the foci, and a new property called eccentricity. We will see that the value of the eccentricity of a conic section can uniquely define that conic.

The three conic sections with their directrices appear in the following figure.

Recall from the definition of a parabola that the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. Therefore, by definition, the eccentricity of a parabola must be 1. The equations of the directrices of a horizontal ellipse are $x=\text{±}\frac{a^2}{c}.$ The right vertex of the ellipse is located at $\left(a,0\right)$ and the right focus is $\left(c,0\right).$ Therefore the distance from the vertex to the focus is $a-c$ and the distance from the vertex to the right directrix is $\frac{a^2}{c}-c.$ This gives the eccentricity as

Since $c<a,$ this step proves that the eccentricity of an ellipse is less than 1. The directrices of a horizontal hyperbola are also located at $x=\text{±}\frac{a^2}{c},$ and a similar calculation shows that the eccentricity of a hyperbola is also $e=\frac{c}{a}.$ However in this case we have $c>a,$ so the eccentricity of a hyperbola is greater than 1.