Problem 60 - 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 \(1+\left(m_{1}+m_{2}\right) \cos w+\) \(m_{1} m_{2}=0\) The axes being inclined at an angle of \(30^{\circ}\), the slope of the line which passes through the point \((-2,3)\) and is perpendicular to the straight line \(y+3 x=6\) is (A) \(\frac{3 \sqrt{3}-2}{\sqrt{3}-6}\) (B) \(\frac{3 \sqrt{3}+2}{\sqrt{3}-6}\) (C) \(\frac{3 \sqrt{3}-2}{\sqrt{3}+6}\) (D) none of these

Step-by-Step Solution

Answer

The slope is \( \frac{3\sqrt{3} - 2}{\sqrt{3} - 6} \). Option (A).

Step 1: Identify Line Equation and Slope

The equation of the line given is in the form of \( y + 3x = 6 \), which can be rewritten as \( y = -3x + 6 \). Thus, the slope \( m_2 \) of this line is \( -3 \).

Step 2: Determine Perpendicular Line Slope Condition

For two lines to be perpendicular, using the given condition: \( 1 + (m_1 + m_2) \cos w + m_1 m_2 = 0 \).

Step 3: Plug Known Values in Perpendicular Condition

Given \( w = 30^{\circ} \), hence \( \cos w = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \). Substitute \( m_2 = -3 \) in the equation: \[ 1 + (m_1 - 3) \frac{\sqrt{3}}{2} + m_1 (-3) = 0 \]

Step 4: Solve for the Slope of Perpendicular Line (m1)

Rearrange and solve \( 1 + (m_1 - 3) \frac{\sqrt{3}}{2} - 3m_1 = 0 \). Simplify to get: \( 1 + \frac{m_1 \sqrt{3}}{2} - \frac{3 \sqrt{3}}{2} - 3m_1 = 0 \). Simplifying further, \[ (\frac{\sqrt{3}}{2} - 3)m_1 = \frac{3 \sqrt{3}}{2} - 1 \]. Solve for \( m_1 \): \[ m_1 = \frac{3\sqrt{3} - 2}{\sqrt{3} - 6} \]

Step 5: Verify Solution

Verify that the calculated \( m_1 = \frac{3\sqrt{3} - 2}{\sqrt{3} - 6} \) matches one of the given options. Check option (A) against our solution.

Key Concepts

Equation of a Line

In the world of mathematics, the equation of a line is a foundation for understanding geometry and algebra. This equation helps to represent lines on a plane, capturing their slope and intercept. The typical form, well-known in Cartesian coordinates, is written as \( y = mx + c \). Here, \( m \) is the slope, showing how steep the line is, and \( c \) is the y-intercept, the point where the line crosses the y-axis. In oblique coordinates, which involve inclined axes (axes that are not perpendicular), the same line gets a new interpretation. The line can appear to rise or fall differently based on the angle that the axes make with one another. Despite this, the linear relationship remains constant, even if expressed differently to accommodate the inclination.

Angle Between Lines

Finding the angle between two lines is crucial in many geometric problems. When two lines intersect, the angle between them can reveal a lot about their relationship. In rectangular coordinates, this angle can be determined using the slopes of the two lines. For example, if lines are represented by equations \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \), the angle \( \theta \) between them can be calculated by the formula: \[ \tan \theta = \frac{|m_1 - m_2|}{1 + m_1m_2} \] However, in oblique coordinates, when axes are inclined, this equation changes. Inclined axes impact how angles are calculated due to the change in reference frames. The modified formula in this scenario is \( \tan \theta = \frac{(m_1 - m_2) \sin w}{1 + (m_1 + m_2) \cos w + m_1m_2} \), focusing on the angle \( w \) between the axes. This alteration accommodates the oblique nature of the canvas we are working with.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's one of the most fundamental concepts in algebra and geometry. Mathematically, the slope is represented by the letter \( m \) and calculated as the rise over run, usually \( m = \frac{\Delta y}{\Delta x} \). A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
When dealing with inclined axes, as mentioned earlier, the slope can be altered by the inclination. In these cases, the perception of steepness changes because the axes themselves are at an angle. This is particularly relevant when determining perpendicular and parallel lines in such a coordinate system. Understanding slopes correctly in these systems is key to solving complex geometry problems efficiently.

Inclined Axes

Inclined axes occur when the x-axis and y-axis are not at a right angle but form some other angle instead. This affects the calculations of line equations and angles between lines, making things a bit tricky.
For example, if two axes form an angle \( w = 30^{\circ} \), this inclination will impact how lines appear and how we compute various properties like slops and angles. It might seem challenging to transition from typical right-angled Cartesian coordinates to an inclined setup, but it opens up a path for understanding more complex and real-world systems.
Inclined axes are used in many advanced fields such as engineering and physics, where systems often do not perfectly align with Cartesian frameworks. Understanding broadens when we learn to adjust our methods to include different axis orientations.