Reflection of Light: Understanding the Incident and Reflected Rays on a Plane Mirror

Introduction: When light rays interact with a reflective surface, they undergo a geometric transformation known as reflection. This phenomenon is governed by the laws of reflection, such as the angle of incidence being equal to the angle of reflection. In this article, we will delve into the mathematical processes and geometric properties involved in determining the equation of the reflected ray when a light ray bounces off a plane mirror.

Understanding the Geometry of Reflection

In the given scenario, we have two lines: an incident ray along the line x - 2y - 3 0, and a plane mirror along the line 3x - 2y - 5 0. When these two lines intersect, the incident ray reflects off the mirror, and we need to find the equation of the reflected ray.

Determining the Point of Incidence

The first step in solving the problem is to find the point of incidence O, where the lines intersect. By setting the equations of the two lines equal to each other:

3x - 2y - 5 0

3x - 6y - 9 0, after multiplying the first equation by 3.

Subtracting the second equation from the first, we get:

4y -4 or y -1, giving us x 1.

Thus, the coordinates of point O are (1, -1).

Calculating the Slopes and Angles

The slope of the incident ray OA can be calculated using the given equation:

x - 2y - 3 0 rearranged to get the slope: y (1/2)x - 3/2.

Hence, the slope mOA 1/2.

The slope of OP (the reflected ray) can be derived similarly from the equation of the mirror: 3x - 2y - 5 0. Rearranging gives y (3/2)x - 5/2, and hence mOP 3/2.

Using the tangent of the angle between two lines, we have:

tan AOP (mOP - mOA) / (1 mOP * mOA)

tan AOP (3/2 - 1/2) / (1 (3/2 * 1/2)) 4/7.

Since AOP COP, the angle AOC is then:

tan AOC 2 * tan AOP 2 * (4/7) / (1 - (4/7)^2) 56/33.

Finding the Slope of the Reflected Ray

The angle formed between the incident ray and the reflected ray can be used to find the slope of the reflected ray. We know that:

tan A 4/7.

From the above calculation, the slope of the reflected ray COB is found to be:

m 29/2.

Thus, the equation of the reflected ray is:

2y 29x - 31 or y (29/2)x - 15.5.

Conclusion and Further Implications

The reflection of light on a plane mirror follows precise geometrical rules and can be determined through algebraic manipulations. Understanding these principles is crucial in physics, engineering, and computer graphics. The above analysis can be applied to a variety of scenarios, including the design of mirrors, the study of optics, and the development of lighting systems.

In summary, the correct equation of the reflected ray is y (29/2)x - 15.5.