A Searchlight

Suppose we need to create a parallel beam of light, as in a searchlight. Where should we place the lightbulb? A parallel beam has zero angle between its rays, so θ_i=0. To place the lightbulb correctly, however, we need to know a distance, not an angle: the distance d_o between the bulb and the mirror. The problem involves a mixture of distances and angles, so we need to get everything in terms of one or the other in order to solve it. Since the goal is to find a distance, let's figure out the image distance corresponding to the given angle θ_i=0. These are related by d_i=1/θ_i, so we have d _i=∞. (Yes, dividing by zero gives infinity. Don't be afraid of infinity. Infinity is a useful problemsolving device.) Solving the distance equation for do, we have

The bulb has to be placed at a distance from the mirror equal to its focal point.