Path length travelled by light - Geometric Optics
Q. A laser is placed at the point . The laser bean travels in a straight line. Hari wants the beam to hit and bounce off the -axis, then hit and bounce off the -axis, then hit the point . What is the total distance the beam will travel along this path?
Every time the laser bounces off a wall, instead we can imagine it going straight by reflecting it about the wall. Thus, the laser starts at and ends at , so the path's length is
Let be the point where the beam hits the -axis, and be the point where the beam hits the -axis.
Reflecting about the -axis gives Then, reflecting about the -axis gives Finally, reflecting about the -axis gives as shown below.
It follows that The total distance that the beam will travel is
Define points and as Solution 2 does.
When a line segment hits and bounces off a coordinate axis at point the ray entering and the ray leaving have negative slopes. Geometrically, these two rays coincide when reflected about the line perpendicular to that coordinate axis, creating line symmetry. Let the slope of be It follows that the slope of is and the slope of is Here, we conclude that
Next, we locate on such that from which is a parallelogram, as shown below.
Let By the property of slopes, we get By symmetry, we obtain
Applying the slope formula on and givesEquating the last two expressions gives
By the Distance Formula, and The total distance that the beam will travel is
Define points and as Solution 2 does.
Since choices and all involve we suspect that one of them is the correct answer. We take a guess in faith that and all form angles with the coordinate axes, from which and The given condition verifies our guess. Following the penultimate paragraph of Solution 3 gives the answer