Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A ray of light is incident along a line which meets another line, \(7 x-y+1=0\), at the point \((0,1)\). The ray is then reflected from this point along the line, \(y+2 x=1\). Then the equation of the line of incidence of the ray of light is : [Online April \(\mathbf{1 0}\), 2016] (a) \(41 \mathrm{x}-25 \mathrm{y}+25=0\) (b) \(41 x+25 y-25=0\) (c) \(41 x-38 y+38=0\) (d) \(41 x+38 y-38=0\)

Step-by-Step Solution

Answer

The equation of the incident line is \(41x - 25y + 25 = 0\). Answer: (a).

Step 1: Verify Intersection Point

Check whether the point \((0,1)\) lies on the line \(7x - y + 1 = 0\). Substitute \(x = 0\) and \(y = 1\) into the equation:\[7(0) - 1 + 1 = 0\]This simplifies to \(0 = 0\), confirming that the point \((0,1)\) is indeed on the line.

Step 2: Determine the Slope of the Reflected Line

Given the equation of the reflected line \(y + 2x = 1\), rewrite it in slope-intercept form:\[y = -2x + 1\]The slope \(m_r\) of the reflected line is \(-2\).

Step 3: Find the Slope of the Normal Line

The original line is \(7x - y + 1 = 0\). Rearrange to find the slope:\[y = 7x + 1\]The slope \(m_o\) of the original line is \(7\). The normal to this line has a slope of \(-\frac{1}{7}\) since the normal slope is the negative reciprocal.

Step 4: Find Slope of Incident Line Using Reflection Property

The incident line, the normal at the point, and the reflected line make equal angles with the normal. Let the slope of the incident line be \(m_i\). Using the reflection property:\[\frac{m_i - (-\frac{1}{7})}{1 + \frac{1}{7}m_i} = \frac{(-2) + \frac{1}{7}}{1 - \frac{1}{7}(-2)}\]Simplify and solve for \(m_i\).

Step 5: Simplify and Solve for Slope of Incident Line

Solve the equation from Step 4:\[\frac{m_i + \frac{1}{7}}{1 + \frac{1}{7}m_i} = \frac{-\frac{13}{7}}{\frac{9}{7}} = -\frac{13}{9}\]Cross-multiplying and simplifying, we solve for \(m_i\) which gives \(m_i = 41/25\).

Step 6: Form the Equation of the Incident Line

With the slope \(m_i = \frac{41}{25}\) and passing through the point \((0,1)\), use point-slope form:\[y - 1 = \frac{41}{25} (x - 0)\]This simplifies to:\[41x - 25y + 25 = 0\]

Step 7: Verify the Equation

Ensure the derived equation matches one of the given options. Notice that the equation \(41x - 25y + 25 = 0\) corresponds to option (a).

Key Concepts

Angle of Incidence

Understanding the angle of incidence is essential when studying the behavior of light. When a light ray hits a surface, the angle of incidence is the angle formed between the incoming ray and an imaginary line that is perpendicular to the surface at the point of contact, known as the "normal".
The angle of incidence plays a crucial role in determining the path of the light after it strikes the surface.

• In our exercise, the ray of light is incident along a line that meets another line, creating a particular angle at the intersection point.
• The unique attribute of this angle is that it helps in predicting the direction of the light's reflection.

By examining the geometry of the problem, the angle between the original ray and the normal is equivalent to the angle of incidence. This symmetrical property is vital for applying the law of reflection, where understanding this angle enables further calculation of subsequent angles and directions involved in the light's path.

Angle of Reflection

The angle of reflection is a fundamental principle in the study of optics. Like the angle of incidence, it is measured from the normal—the line perpendicular to the surface where the light ray reflects. According to the law of reflection, the angle of reflection is always equal to the angle of incidence.
This principle provides predictability in optical paths and is a cornerstone idea in both physics and related applications.

• In the given problem, once the light hits the intersection point, it reflects off into a new direction defined by another line.
• This reflection adheres to the law of reflection, ensuring that the light's path can be traced and calculated accurately.

Using the equality of the angles of incidence and reflection simplifies the analysis of light behavior in complex scenarios. It allows engineers and scientists to design systems, such as mirrors and optics, that precisely control light direction.

Equation of a Line

The equation of a line in a two-dimensional space is a foundational concept in geometry and algebra. It provides a mathematical way to describe straight lines using an equation that typically takes the slope-intercept form: \[ y = mx + c \] where \(m\) represents the slope and \(c\) the y-intercept.
In various contexts, the equation can also be represented in point-slope form \( (y - y_1) = m(x - x_1) \) or in general form \( Ax + By + C = 0 \).

• The slope \(m\) describes the line's steepness and direction. A positive slope indicates an upward rising line, while a negative slope indicates a downward slope.
• In our exercise, determining the equation for both the incident and reflected lines helps trace the light’s complete path.

Starting from a point known to lie on the line and with the calculated slope, one can apply these forms to figure out the exact line equation. This exercise illustrates using the line equation to analyze and solve a practical problem involving angles and movement, showcasing its versatility in applications beyond pure mathematics.