Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the straight lines \(y=m_{1} x+c_{1}\) and \(y=m_{2} x+c_{2}\) make equal angles with the axis of \(x\) and be not parallel to one another, then \(m_{1}+m_{2}+k m_{1} m_{2} \cos w=0\) where \(k=\) (A) 1 (B) 2 (C) \(-1\) (D) \(-2\)

Step-by-Step Solution

Answer

The value of \(k\) is (A) 1.

Step 1: Understand the condition on angles with the x-axis

Two lines making equal angles with the x-axis means the slopes, \(m_1\) and \(m_2\), are such that \(|\tan^{-1}(m_1)| = |\tan^{-1}(m_2)|\). Therefore, \(m_1 = -m_2\) or \(m_1 = m_2\). Since the lines are not parallel, \(m_1 = -m_2\).

Step 2: Use the condition provided in the problem

Since it is given that \(m_1 = -m_2\), substitute to verify the condition \(m_1 + m_2 + k m_1 m_2 \cos w = 0\).

Step 3: Simplify the equation

Since \(m_1 + m_2 = 0\) (from \(m_1 = -m_2\)), the equation simplifies to \(k m_1 m_2 \cos w = 0\). This implies \(k = 0\) or \(\cos w = 0\).

Step 4: Determine the independent term

Since the problem provides an independent coefficient \(k\) among choices that are not zero, use the condition that \(\cos w = 0\) to find the relation of \(k\) with \(m_1\) and \(m_2\). Given this, \(k m_1 m_2 = 0\), substituting \(-m_2\) gives the relation \(1 + k(-1) = 0\).

Step 5: Solve for the value of k

Solve the equation \(1 - k = 0\) by finding \(k = 1\). Therefore, the correct value of \(k\) is \(1\).

Key Concepts

Slopes of Lines

The slope of a line in coordinate geometry is a measure of its steepness. When you look at the equation of a line in the form of \(y = mx + c\), \(m\) represents the slope. A slope value tells how much \(y\) changes for a unit change in \(x\).
In the context of our exercise, the slopes \(m_1\) and \(m_2\) are crucial. They dictate the angles each line makes with the x-axis. When two lines make equal angles with the x-axis, their slopes are related in a specific way.

• If the slopes \(m_1\) and \(m_2\) have the same magnitude but opposite signs, it means the lines are perpendicular but not parallel. Hence, \(m_1 = -m_2\).
• This property of the slope allows us to interpret the geometric orientation of a line, which is fascinating in coordinate geometry.

Angles with Axes

Understanding the angle a line makes with the axes is essential in geometry. The angle a line makes with the x-axis can be understood using its slope, \(m\). The angle \(\theta\) is given by \(\theta = \tan^{-1}(m)\).
In cases where two lines make equal angles with the x-axis, it implies the absolute values of their angles are the same. However, this equality is achieved under two main conditions: either the angles are equal or supplementary.

• In our exercise, since \(m_1 = -m_2\), the lines have slopes with equal angles but of opposite direction with respect to the x-axis.
• Such situations frequently occur in problems involving coordinate lines and involve an understanding that, geometrically, angles can "balance" each other out because of these properties.

Trigonometric Identities

Trigonometric identities often play a key role in simplifying expressions involving angles and slopes. In this exercise, cosine plays a significant role, which is a basic trigonometric function.
The identity used here is related to cosine. Specifically, \(\cos w\) was central to reducing our equation under the conditions \(m_1 + m_2 + k m_1 m_2 \cos w = 0\).

• Because \(\cos w = 0\) was explored as a condition, knowing \(\cos 90^\circ = 0\), it implies the angle between the lines and the x-axis was crucially involved.
• This simplification leads to additional deductions about the variable \(k\), guiding us efficiently to conclude that \(k = 1\).

Understanding these interdependencies, especially how trigonometric identities like cosine simplify equations, provides a deeper insight into coordinate geometry problems.