Creating a Hyperbola with a Flashlight

Question

I ran into this problem in a textbook and was intrigued by it. Conics are generally formed through different cuts one can make with the shape of a cone. But, there have been recent discussions on creating conical shapes through aiming a flashlight in a certain direction, which is interesting.

If you hold a flashlight parallel to the ground, the beam would be able to create a parabolic shape on the ground, which is one of the three conical shapes that can be made through a flashlight. Similarly, if one tilts a flashlight by aiming it on the wall, it could create an elliptical shape. However, I'm uncertain about what one can do to create a hyperbolic shape with a flashlight.

This picture is the reference for this problem and asks two things which I'm wondering if anyone has input on: (a) Why is the boundary of this lighted area a hyperbola? (b) How can one hold a flashlight so that its beam forms a hyperbola on the ground?

Answer 1

This picture might be a good supplement to Neal's explanation, which is spot-on. If you imagine that your flashlight is pointing straight down from the tip of the cone, then the wall is any one of the colored areas, depending on the angle you hold it at.

Answer 2

Start with the flashlight parallel to the wall. If you incline it slightly away from the wall, the (conical) beam will still hit the wall and you'll get a hyperbola. It is only after the angle of the flashlight to the wall is greater than the angle of the light cone that the hyperbola vanishes at infinity. (In practice, of course, it is rather before then because the air will scatter the light.)

As CiaPan pointed out in the comments below, the criterion for being a parabola is that the cut be parallel to the opposite side of the light cone. The axis of the cone is the direction of the flashlight. Therefore you still get a hyperbola for the flashlight inclined toward the wall at shallow angles.