Problem 29 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(l_{1}, m_{1}, n_{1}\) and \(l_{2}, m_{2}, n_{2}\) are d.c.'s of the two lines inclined to each other at an angle \(\theta\), then the d.c.'s of the internal bisector of the angle between these lines are (A) \(\frac{l_{1}+l_{2}}{2 \sin \theta / 2}, \frac{m_{1}+m_{2}}{2 \sin \theta / 2}, \frac{n_{1}+n_{2}}{2 \sin \theta / 2}\) (B) \(\frac{l_{1}+l_{2}}{2 \cos \theta / 2}, \frac{m_{1}+m_{2}}{2 \cos \theta / 2}, \frac{n_{1}+n_{2}}{2 \cos \theta / 2}\) (C) \(\frac{l_{1}-l_{2}}{2 \sin \theta / 2}, \frac{m_{1}-m_{2}}{2 \sin \theta / 2}, \frac{n_{1}-n_{2}}{2 \sin \theta / 2}\) (D) \(\frac{l_{1}-l_{2}}{2 \cos \theta / 2}, \frac{m_{1}-m_{2}}{2 \cos \theta / 2}, \frac{n_{1}-n_{2}}{2 \cos \theta / 2}\)

Step-by-Step Solution

Answer

The correct option is (B).

Step 1: Understanding the directional cosines (d.c.'s)

The directional cosines (d.c.'s) of a line are the cosines of the angles the line makes with the coordinate axes, represented as \(l, m, n\). For two lines with d.c.'s \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the angle \(\theta\) between them is given by \(\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\).

Step 2: Using the formula for d.c.'s of the angle bisector

If two lines have d.c.'s \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the d.c.'s of the internal bisector of the angle between these lines are given by the formula: \(\frac{l_1 + l_2}{2 \cos \theta / 2}, \frac{m_1 + m_2}{2 \cos \theta / 2}, \frac{n_1 + n_2}{2 \cos \theta / 2}\).

Step 3: Identify the correct option

From the options provided, option (B) matches the formula for the d.c.'s of the internal bisector of the angle between the lines. It is \(\frac{l_1 + l_2}{2 \cos \theta / 2}, \frac{m_1 + m_2}{2 \cos \theta / 2}, \frac{n_1 + n_2}{2 \cos \theta / 2}\).

Key Concepts

Angle Between Lines

Understanding the angle between lines is crucial for solving problems involving directional cosines and internal bisectors. When two lines are inclined to each other, we can find the angle between them using their directional cosines. Directional cosines are denoted as \(l, m, n\), which are essentially the cosines of the angles between the line and the coordinate axes (x, y, z). For two lines, these are represented as \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\). To find the angle \(\theta\) between these lines, we use the formula:\[\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\]This formula calculates the dot product of the directional vectors of the two lines; it offers a straightforward pathway to figure out \(\theta\). Since cosines range between -1 and 1, this helps determine how the lines relate—which ensures we understand whether they are acute or obtuse to each other, giving insight into the angle's nature.

Internal Bisector

The internal bisector of an angle between two lines is a line that divides the angle into two equal parts. In mathematical terms, if we have two lines with directional cosines \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\), the internal bisector's directional cosines can be determined using the following formula:\[\left( \frac{l_1 + l_2}{2 \cos \theta / 2}, \frac{m_1 + m_2}{2 \cos \theta / 2}, \frac{n_1 + n_2}{2 \cos \theta / 2} \right)\]The \(\cos \theta/2\) aspect of the formula highlights that we consider the cosine of half the angle between the two lines. This decision emphasizes symmetry as it creates equal partitions, transforming complex visual geometry into approachable mathematics. Remember that the internal bisector is unique to the angle's type; accurate calculations depend on recognizing and applying this diligently. Good understanding of this concept ensures efficient problem-solving in more extensive geometry tasks.

JEE Main Mathematics

JEE Main is a competitive examination for students aspiring to enter engineering colleges in India. The mathematics section of this exam includes numerous topics, such as coordinate geometry, which are crucial for scoring well. Understanding directional cosines and internal bisectors is directly relevant to this area. Problems may present lines with given directional cosines and ask students to find the angle between them, or the directional cosines of an angle bisector. For a successful JEE Main preparation, students should focus on:

• Conceptual clarity: Grasping the underlying geometry
• Practice: Solving a variety of problems builds confidence
• Efficiency: Quickly applying formulas to maximize score

Mastery of these principles not only aids in examination performance but also forms a foundation for advanced studies in fields like engineering and physics. By delving deeper into such concepts, students build a robust mathematical skill set that will benefit them in the long term.