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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.

A hyperbola is drawn that is open to the right and left. There is a ray pointing to a point on the right hyperbola marked “Light from star.” It hits a “Mirror surface” and bounces to the focus on the other side of the hyperbola. There is dashed line from where the point hits the mirror surface to the focus on that side of the hyperbola.
A hyperbolic mirror used to collect light from distant stars.

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.

Definition

The eccentricity     e of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. This value is constant for any conic section, and can define the conic section as well:

  1. If e = 1 , the conic is a parabola.
  2. If e < 1 , it is an ellipse.
  3. If e > 1 , it is a hyperbola.

The eccentricity of a circle is zero. The directrix of a conic section is the line that, together with the point known as the focus, serves to define a conic section. Hyperbolas and noncircular ellipses have two foci and two associated directrices. Parabolas have one focus and one directrix.

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

This figure has three figures. In the first is an ellipse, with center at the origin, foci at (c, 0) and (−c, 0), half of its vertical height being b, half of its horizontal length being a, and directrix x = ±a2/c. The second figure is a parabola with vertex at the origin, focus (a, 0), and directrix x = −a. The third figure is a hyperbola with center at the origin, foci at (c, 0) and (−c, 0), vertices at (a, 0) and (−a, 0), and directices at x = ±a2/c.
The three conic sections with their foci and directrices.

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 = ± a 2 c . The right vertex of the ellipse is located at ( a , 0 ) and the right focus is ( c , 0 ) . Therefore the distance from the vertex to the focus is a c and the distance from the vertex to the right directrix is a 2 c c . This gives the eccentricity as

e = a c a 2 c a = c ( a c ) a 2 a c = c ( a c ) a ( a c ) = c a .

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 = ± a 2 c , and a similar calculation shows that the eccentricity of a hyperbola is also e = c a . However in this case we have c > a , so the eccentricity of a hyperbola is greater than 1.

Determining eccentricity of a conic section

Determine the eccentricity of the ellipse described by the equation

( x 3 ) 2 16 + ( y + 2 ) 2 25 = 1 .

From the equation we see that a = 5 and b = 4 . The value of c can be calculated using the equation a 2 = b 2 + c 2 for an ellipse. Substituting the values of a and b and solving for c gives c = 3 . Therefore the eccentricity of the ellipse is e = c a = 3 5 = 0.6 .

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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