A light situated at a point in a plane sends out beams of light in all directions. The bea...

A light situated at a point in a plane sends out beams of light in all directions. The beams in the plane meet a curve and are all reflected parallel to a line in the plane, as shown in Figure.

Figure. The reflector in Exercise.

The light is reflected so that the angle of incidence αequals the angle of reflection ß.

(a) Show that tanθ = tan2ß; then use trigonometry to show that

.

(b) Use the quadratic formula to solve equation; then solve the resulting first-order differential equation to find the equation of the reflecting curve. Hint: You may want to try some of Exercises before attempting this solution.