Begin2.DVI

Using algebra one can verify that the equation (7.7) reduces down to an identity so

that the angles αand βare equal. This in turn implies θ 1 =θ 2 which states that the

angle of incidence equals the angle of reflection.

Sound waves also are reflected in the same way. Elliptically shaped rooms or

domes have the property that someone whispering at one focus of the ellipse can

easily be heard at the other focus of the ellipse. This gives rise to the phrase

“whispering galleries”. Constructions which make use of this reflection property

of the ellipse can be found at Statuary Hall in the United States capital, St Paul’s

Cathedral in London, the Grand Central Terminal in New York City and in certain

museums throughout the world.

Example 7-3. Reflection property of the hyperbola.

Figure 7-3.

Light ray directed toward one focus reflects and goes through other focus.

Sketch the hyperbola

x^2

a^2 −

y^2

b^2 = 1 with foci F^1 , F^2 having coordinates (−c,0)

and (c,0) respectively, where c=ae and e > 1 is the eccentricity of the hyperbola.

Construct an incident ray aimed at the focus F 2 which passes through a known point

(x 0 , y 0 )on the hyperbola. Label the point where this ray intersects the hyperbola

as point P. Sketch the tangent line and normal line to the hyperbola at the point

P and label the angle of incidence as θ 1 and the angle of reflection as θ 2. The