The reflective surface is given by y=2 sinx . The reflective...

The reflective surface is given by $y=2 \sin x .$ The reflective surface is facing positive axis. What is the least values of co-ordinate of the point where a ray parallel to positive $x$ -axis becomes parallel to positive $y$ -axis after reflection? (a) $\left(\frac{\pi}{3}, \sqrt{3}\right)$ (b) $\left(\frac{\pi}{2}, \sqrt{2}\right)$ (c) $\left(\frac{\pi}{3}, \sqrt{2}\right)$ (d) $\left(\frac{\pi}{4}, \sqrt{3}\right)$

The reflective surface is given by $y=2 \sin x .$ The reflective surface is facing positive axis. What is the least values of co-ordinate of the point where a ray parallel to positive $x$ -axis becomes parallel to positive $y$ -axis after reflection?
(a) $\left(\frac{\pi}{3}, \sqrt{3}\right)$
(b) $\left(\frac{\pi}{2}, \sqrt{2}\right)$
(c) $\left(\frac{\pi}{3}, \sqrt{2}\right)$
(d) $\left(\frac{\pi}{4}, \sqrt{3}\right)$

Key Concepts

Optimization in Geometric Problems

When a problem asks for the 'least' coordinate or minimal point fulfilling a given condition, it requires not only setting up the necessary geometric relationships (such as those from the law of reflection) but also selecting the solution that minimizes the coordinate value. This involves solving equations derived from the reflection conditions and then choosing the solution that meets any additional criteria such as being the smallest in a specified interval or domain.

Vector Components and Angle Bisectors

The analysis of reflection often involves representing directions as vectors. The incident and reflected rays are treated as vectors, and their relation through the normal involves the concept of angle bisectors. In many reflection problems, the normal vector is characterized by being proportional to the sum of the incident and reflected unit vectors. This vector approach aids in setting up the equations that enforce the law of reflection.

Differentiation and Slope

Differentiation is used to compute the slope of the tangent line to a curve at any given point. This slope, given by the derivative, is essential for finding the orientation of the tangent line and by extension the normal vector. In the context of reflection from a curved surface, finding the derivative at the point of incidence sets up the equation needed to satisfy the reflection condition.

Normal and Tangent Lines

For any smooth curve, the tangent line at a point represents the instantaneous direction of the curve, and the normal line is the line perpendicular to the tangent at that point. In reflection problems, the normal line is used to measure the angles of incidence and reflection. Determining the correct normal vector from the derivative of the curve allows one to apply the law of reflection correctly, especially when the reflective side of the surface is specified.

Law of Reflection

This principle states that when a ray of light reflects off a surface, the angle of incidence (the angle between the incoming ray and the normal to the surface) is equal to the angle of reflection (the angle between the reflected ray and the normal). This law is central to determining how a ray changes direction upon striking a reflective surface, and it is fundamental in solving problems where the direction of the incident and reflected rays (such as horizontal and vertical directions) are specified.

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