Problem 61 - A Complete Resource Book in Mathematics for JEE Main

Question

In oblique coordinates, the equation \(y=m x+c\) represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the \(x\)-axis, where \(w\) is the angle between the axes. If \(\theta\) be the angle between two lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x\) \(+c_{2}, w\) be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if \(m_{1}=m_{2}\). The two lines are perpendicular if \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) If \(y=x \tan \frac{11 \pi}{24}\) and \(y=x \tan \frac{19 \pi}{24}\) represent two straight lines at right angles, then the angle between the axes is (A) \(\frac{\pi}{6}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{3}\) (D) \(\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The angle between the axes is \(\frac{\pi}{6}\) (option A).

Step 1: Identify Given Equations and Their Slopes

The two lines given are \(y = x \tan \frac{11\pi}{24}\) and \(y = x \tan \frac{19\pi}{24}\). Therefore, the slopes of the lines \(m_1 = \tan \frac{11\pi}{24}\) and \(m_2 = \tan \frac{19\pi}{24}\).

Step 2: Determine the Condition for Perpendicular Lines

For the two lines to be perpendicular, the condition \(1 + (m_1 + m_2) \cos w + m_1 m_2 = 0\) must hold. Here, \(w\) is the angle between the axes.

Step 3: Calculate \(m_1 + m_2\) and \(m_1 m_2\)

Calculate \(m_1 + m_2 = \tan \frac{11\pi}{24} + \tan \frac{19\pi}{24}\) and \(m_1 m_2 = \tan \frac{11\pi}{24} \cdot \tan \frac{19\pi}{24}\).

Step 4: Simplify the Perpendicular Condition Equation

Substitute the calculated values of \(m_1 + m_2\) and \(m_1 m_2\) into the perpendicular condition equation: \(1 + (m_1 + m_2) \cos w + m_1 m_2 = 0\).

Step 5: Solve for \(w\)

Rearrange the equation to solve for \(\cos w\). Simplify and compute the possible value(s) for \(w\).

Step 6: Interpret the Result

Verify which of the solution options corresponds to the value of \(w\).

Key Concepts

Equation of Straight Line

In oblique coordinates, straight lines behave similarly to how they do in traditional Cartesian coordinates. However, these lines are represented concerning a non-right angle between the axes, denoted by an angle \(w\). In general, a line is defined by the equation \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept.

- **Slope \(m\):** Describes the line's steepness and is a critical factor in line-related calculations, like angle determination.
- **Y-intercept \(c\):** The point where the line meets the y-axis.

The slope \(m\) can be recalculated using trigonometric modifications when considering oblique axes. Understanding how to form and manipulate straight line equations in this context is foundational for solving more complex problems that involve these lines.

Angle Between Axes

In oblique coordinate systems, the axes are not necessarily perpendicular. The angle between the axes is essential because it alters how we calculate angles and distances. This angle is denoted by \(w\) in problems involving lines and determines how the slopes influence each other.

Calculating the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is adapted using trigonometry in such systems. The specific formula is: \[\tan \theta = \frac{(m_1 - m_2) \sin w}{1 + (m_1 + m_2) \cos w + m_1 m_2}\]

This formula adjusts the typical tangent calculation to include the angle \(w\) between the axes, providing a method to measure \(\theta\) based on modified slopes. As you explore problems, understanding this adaptation becomes crucial.

Perpendicular Lines

Perpendicular lines in oblique coordinates present a unique challenge since the relationship is not as straightforward as a 90-degree angle in Cartesian systems. Instead, we use a specific condition to assess perpendicularly.

For lines \(y = m_1 x + c_1\) and \(y = m_2 x + c_2\) to be perpendicular in an oblique framework, the following condition must be met: \[1 + (m_1 + m_2) \cos w + m_1 m_2 = 0\]

Here, \(w\) is the angle between the oblique axes. Solving this can provide insights into the spatial configuration and lead to the conclusion of perpendicularity in functional terms. In practice problems, verifying perpendicularity involves substituting the given slopes and determining if this condition holds.

Trigonometric Functions

Trigonometric functions help manage the complexities associated with oblique coordinates. They enable transformations of standard geometric relationships by considering additional angles. Basic functions like sine and cosine are imperative when calculating aspects like angles between lines.

In oblique systems, trigonometry is used to adjust calculations affected by the axis angle \(w\). For example, the angle \(\theta\) between two lines relates directly to \(\tan^{-1}\), combining trig functions to accommodate the skewed axes: \[\theta = \tan^{-1}\left(\frac{m \sin w}{1 + m \cos w}\right)\]

This formula emphasizes the role of trigonometry in converting classic Euclidean geometry to solve real-world problems in diverse coordinate systems. Mastery of these trigonometric applications is crucial for anyone dealing with oblique axes.