# 1.5 Conic sections  (Page 5/23)

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Ellipses also have interesting reflective properties: A light ray emanating from one focus passes through the other focus after mirror reflection in the ellipse. The same thing occurs with a sound wave as well. The National Statuary Hall in the U.S. Capitol in Washington, DC, is a famous room in an elliptical shape as shown in [link] (b). This hall served as the meeting place for the U.S. House of Representatives for almost fifty years. The location of the two foci of this semi-elliptical room are clearly identified by marks on the floor, and even if the room is full of visitors, when two people stand on these spots and speak to each other, they can hear each other much more clearly than they can hear someone standing close by. Legend has it that John Quincy Adams had his desk located on one of the foci and was able to eavesdrop on everyone else in the House without ever needing to stand. Although this makes a good story, it is unlikely to be true, because the original ceiling produced so many echoes that the entire room had to be hung with carpets to dampen the noise. The ceiling was rebuilt in 1902 and only then did the now-famous whispering effect emerge. Another famous whispering gallery—the site of many marriage proposals—is in Grand Central Station in New York City.

## Hyperbolas

A hyperbola can also be defined in terms of distances. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes.

## Definition

A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant.

A graph of a typical hyperbola appears as follows.

The derivation of the equation of a hyperbola in standard form is virtually identical to that of an ellipse. One slight hitch lies in the definition: The difference between two numbers is always positive. Let P be a point on the hyperbola with coordinates $\left(x,y\right).$ Then the definition of the hyperbola gives $|d\left(P,{F}_{1}\right)-d\left(P,{F}_{2}\right)|=\text{constant}.$ To simplify the derivation, assume that P is on the right branch of the hyperbola, so the absolute value bars drop. If it is on the left branch, then the subtraction is reversed. The vertex of the right branch has coordinates $\left(a,0\right),$ so

$d\left(P,{F}_{1}\right)-d\left(P,{F}_{2}\right)=\left(c+a\right)-\left(c-a\right)=2a.$

This equation is therefore true for any point on the hyperbola. Returning to the coordinates $\left(x,y\right)$ for P :

$\begin{array}{ccc}\hfill d\left(P,{F}_{1}\right)-d\left(P,{F}_{2}\right)& =\hfill & 2a\hfill \\ \hfill \sqrt{{\left(x+c\right)}^{2}+{y}^{2}}-\sqrt{{\left(x-c\right)}^{2}+{y}^{2}}& =\hfill & 2a.\hfill \end{array}$

$\begin{array}{ccc}\hfill \sqrt{{\left(x-c\right)}^{2}+{y}^{2}}& =\hfill & 2a+\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}\hfill \\ \hfill {\left(x-c\right)}^{2}+{y}^{2}& =\hfill & 4{a}^{2}+4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+{\left(x+c\right)}^{2}+{y}^{2}\hfill \\ \hfill {x}^{2}-2cx+{c}^{2}+{y}^{2}& =\hfill & 4{a}^{2}+4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+{x}^{2}+2cx+{c}^{2}+{y}^{2}\hfill \\ \hfill \text{−}2cx& =\hfill & 4{a}^{2}+4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+2cx.\hfill \end{array}$

Now isolate the radical on the right-hand side and square again:

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