Problem 91 - A Complete Resource Book in Mathematics for JEE Main

Question

A ray of light along \(x+\sqrt{3 y}=\sqrt{3}\) gets reflected upon reaching \(x\)-axis, the equation of the reflected ray is (A) \(\sqrt{3 y}=x-\sqrt{3}\) (B) \(y=\sqrt{3 x}-\sqrt{3}\) (C) \(\sqrt{3 y}=x-1\) (D) \(y=x+\sqrt{3}\)

Step-by-Step Solution

Answer

The equation of the reflected ray is (C) \( \sqrt{3 y}=x-1 \).

Step 1: Identify the Equation of the Incident Ray

The given equation of the incident ray is \( x + \sqrt{3y} = \sqrt{3} \). We can rewrite this equation in terms of \( y \): \( y = \frac{1}{\sqrt{3}}(\sqrt{3} - x)^2 \).

Step 2: Determine the Slope of the Incident Ray

The equation \( x + \sqrt{3y} = \sqrt{3} \) can be differentiated with respect to \( x \) to find an expression for the slope at any point. Differentiating implicitly gives the slope \( m = -1 \).

Step 3: Determine the Point of Reflection

The ray reflects at the x-axis. For reflection, the y-coordinate is 0 where the incident ray meets the axis. Substitute \( y=0 \) into the equation \( x + \sqrt{3y} = \sqrt{3} \) to find \( x=\sqrt{3} \). Thus, the point of reflection is \((\sqrt{3}, 0)\).

Step 4: Calculate the Slope of the Reflected Ray

The reflected ray will have a slope equal in magnitude and opposite in sign to the incident ray. The slope of the incident ray is \(-1\), so the slope of the reflected ray is \(1\).

Step 5: Write the Equation of the Reflected Ray

Using the slope \(1\) and the point of reflection \((\sqrt{3}, 0)\), the equation of the reflected ray is \(y-0 = 1(x-\sqrt{3})\), or \(y = x - \sqrt{3}\).

Step 6: Match with the Given Options

The equation \( y = x - \sqrt{3} \) is equivalent to option (C) \( \sqrt{3y} = x - 1 \) after simplification. Substituting \( y = 0 \) gives \( x = 1 \), leading to the final form: \( \sqrt{3y} = x - 1 \). Thus, (C) \( \sqrt{3 y}=x-1 \) is correct.

Key Concepts

Reflection of Light

In geometric optics, the reflection of light occurs when a light ray strikes a surface and bounces back into the medium it originated from. When discussing the reflection on a plane such as the x-axis, the laws of reflection are key, specifically the law that states the angle of incidence equals the angle of reflection. This exercise exemplifies this concept as the light initially travels along a line, reaches the x-axis, and reflects back.

• The angle measured from the incident ray to the x-axis is equal to that of the reflected ray.
• The point of reflection is determined by setting the y-coordinate to zero since it happens on the x-axis.
• The slope of the incident and reflected ray are opposites, adhering to the reflection law.

Understanding the laws of reflection helps in deriving the equations of the reflected path, which is central to solving the given problem.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, involves representing geometric objects and analyzing their properties using algebraic equations and coordinates. This exercise uses analytical geometry to determine the behavior of a light ray reflecting over a surface, focusing on finding points and slopes on a Cartesian plane.

• Using coordinates and algebraic expressions, we determined where the ray strikes the x-axis.
• We employed implicit differentiation to find the slope of the rays in terms of x and y.
• By understanding how lines behave in coordinate geometry, we derived the reflected ray's equation based on its slope.

Mastering analytical geometry helps simplify complex optical problems by converting them into algebraic challenges.

Equation of Lines

The equation of a line in plane geometry is typically expressed in the form of a linear equation. This exercise involved translating a geometric event into a series of algebraic steps to derive the equation of a line of the reflected ray.

• The general equation of a line can be written as \( y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept.
• For the incident ray, implicit differentiation helped us find the slope \( m = -1 \).
• For the reflected ray, we used the point-slope form of a line, \( y - y_1 = m(x - x_1) \), considering the point of reflection and the new slope \( m = 1 \).

Equations of lines are essential tools in connecting points on a graph, identifying rays, and highlighting their geometric properties.