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

Question

A line in the 3-dimensional space makes an angle \(\theta\) \(\left(0<\theta \leq \frac{\pi}{2}\right)\) with both the \(x\) and y axes. Then the set of all values of \(\theta\) is the interval: \(\quad\) [Online April 9, 2014] (a) \(\left(0, \frac{\pi}{4}\right]\) (b) \(\left[\frac{\pi}{6}, \frac{\pi}{3}\right]\)(c) \(\left[\frac{\pi}{4}, \frac{\pi}{2}\right]\) (d) \(\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\)

Step-by-Step Solution

Answer

The interval is \([\frac{\pi}{4}, \frac{\pi}{2}]\), which is option (c).

Step 1: Understanding the Problem

We need to find the range of possible angles \( \theta \) such that a line in 3D space makes the same angle with both the \( x \) and \( y \) axes. \( \theta \) satisfies \( 0 < \theta \leq \frac{\pi}{2} \). We must determine which given interval describes this range of \( \theta \).

Step 2: Analyzing the Condition

Consider the condition of the problem: A line makes the same angle \( \theta \) with the \( x \) and \( y \) axes. In 3D geometry, the condition for a line making angles with all coordinate axes is determined by the direction cosines, \( \cos \theta_x, \cos \theta_y, \cos \theta_z \), and must satisfy:\[\cos^2 \theta_x + \cos^2 \theta_y + \cos^2 \theta_z = 1.\]

Step 3: Equating Angles with Two Axes

Since \( \theta_x = \theta_y = \theta \), this simplifies to \( \cos^2 \theta + \cos^2 \theta + \cos^2 \theta_z = 1 \), or \( 2 \cos^2 \theta + \cos^2 \theta_z = 1 \). This means \( \cos^2 \theta \leq \frac{1}{2} \).

Step 4: Solving the Trigonometric Inequality

Solve \( \cos^2 \theta \leq \frac{1}{2} \). This leads to:\[ |\cos \theta| \leq \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}. \]Since \( \theta \) is between 0 and \( \frac{\pi}{2} \), \( \cos \theta \) should be non-negative. So:\[ \cos \theta \leq \frac{\sqrt{2}}{2}. \]

Step 5: Finding Range for \( \theta \)

Using the inverse cosine function, when \( \cos \theta \leq \frac{\sqrt{2}}{2} \), we have \( \theta \geq \frac{\pi}{4} \). Therefore, the interval for \( \theta \) that satisfies the condition is \( \left[ \frac{\pi}{4}, \frac{\pi}{2} \right] \).

Step 6: Verifying the Solution

The solution interval \( \left[ \frac{\pi}{4}, \frac{\pi}{2} \right] \) aligns with answer choice (c). This interval represents the range of angles where the line in 3D space makes an equal angle with both the \( x \) and \( y \) axes, up to \( \frac{\pi}{2} \).

Key Concepts

Direction Cosines

In 3D geometry, direction cosines play a crucial role when dealing with lines and their orientation in space. Direction cosines refer to the cosines of the angles that a specific line makes with each of the coordinate axes. For any line in three-dimensional space, the direction cosines are denoted as \( \cos \theta_x \), \( \cos \theta_y \), and \( \cos \theta_z \). These correspond to the angles that the line makes with the x-axis, y-axis, and z-axis, respectively.

The fundamental property of direction cosines is that they satisfy the equation \( \cos^2 \theta_x + \cos^2 \theta_y + \cos^2 \theta_z = 1 \). This equation is derived from the Pythagorean identity in three dimensions and ensures that the concept of direction cosines is consistent with the physics of space.

When a line makes equal angles with two axes, say the x and y axes, it implies that \( \theta_x = \theta_y = \theta \). In this scenario, our equation simplifies as follows:

• \( 2\cos^2 \theta + \cos^2 \theta_z = 1 \).

For students wrestling with the idea of direction cosines, remember:

• They show how a line is oriented concerning coordinate axes.
• They always add up to unity in their squared form.

Trigonometric Inequality

Trigonometric inequalities, such as the one presented when solving for \( \theta \) in 3D geometry problems, require understanding how trigonometric functions behave within certain bounds. The inequality we encounter here is \( \cos^2 \theta \leq \frac{1}{2} \).

The goal is to determine the range of angle \( \theta \) where this inequality holds true. Since the base of our trigonometric function, \( \cos^2 \theta \), represents a cosine angle squared, we start by manipulating:

• The square root inequality: \( |\cos \theta| \leq \frac{1}{\sqrt{2}} \).

Recognizing that \( \theta \) ranges from 0 to \( \frac{\pi}{2} \) eliminates negative values of \( \cos \theta \):

• Thus, \( \cos \theta \leq \frac{\sqrt{2}}{2} \).

Once you solve \( \cos \theta \leq \frac{\sqrt{2}}{2} \), inversely applying the cosine function leads us to determine the angle \( \theta \geq \frac{\pi}{4} \). Keeping all trigonometric inequalities within a specific range allows you to find the correct interval of solutions.

Angles in 3D Space

Understanding angles in 3D space is critical for analyzing the geometry of solid shapes and lines intersecting planes at various orientations. When a problem states that a line makes the same angle with both the x and y axes in 3D space, it assumes a T-formed plane where this equal angle is key to further solutions.

In typical scenarios, like the problem exercise, knowing that an angle \( \theta \) with multiple axes requires examining how such angles interplay. Using direction cosines and trigonometric inequalities, one ensures that angles in 3D do not casually overlap or contradict set mathematical principles.

For the interval of \( \theta \), when a line is at these equal angles referred to earlier, a geometric understanding confirms that within \( \left[ \frac{\pi}{4}, \frac{\pi}{2} \right] \), all conditions that align with direction cosine principles must stay intact:

• The line can be visualized slicing through a 3D plane portion between the boundary angles \( \frac{\pi}{4} \) and \( \frac{\pi}{2} \).
• Thus forming segments and vectors in xyz space where these condition-specific angles perfectly balance.

With these insights, students will better comprehend how lines, planes, and vectors coexist harmoniously and the importance of visualizing such geometrical relationships.