Problem 38 - A Complete Resource Book in Mathematics for JEE Main

Question

Due south of a tower which is leaning towards north there are two stations at distances \(x\) and \(y\) respectively from its foot. If \(\alpha, \beta\) respectively be the angles of elevation of the top of the tower at these stations, then the inclination \(\theta\) of the tower to the horizontal is given by \(\cot \theta=\) (A) \(\frac{y \cot \alpha-x \cot \beta}{y-x}\) (B) \(\frac{y \cot \alpha+x \cot \beta}{y-x}\) (C) \(\frac{y \cot \alpha-x \cot \beta}{y+x}\) (D) \(\frac{y \tan \alpha-x \sin \beta}{y-x}\)

Step-by-Step Solution

Answer

Option (A): \( \frac{y \cot \alpha - x \cot \beta}{y - x} \) is correct.

Step 1: Understanding the Problem

We have two stations south of a tilted tower, located at distances \(x\) and \(y\). The angles of elevation to the top of the tower from these stations are \(\alpha\) and \(\beta\) respectively. We need to find \(\cot \theta\), where \(\theta\) is the angle of inclination of the tower.

Step 2: Determine the Height of the Tower

The height of the tower does not change, and it is used to define the angles \(\alpha\) and \(\beta\). From horizontal distances \(x\) and \(y\), the height \(h\) of the tower is given by:1. \( h = x \tan \alpha \)2. \( h = y \tan \beta \)

Step 3: Relate Height with Cotangent Formulas

Based on the \(h = x \tan \alpha\) and \(h = y \tan \beta\) equations, solve for \(h\):\( x \tan \alpha = y \tan \beta = h \)Since we need to express \(\tan \alpha\) and \(\tan \beta\) in terms of \(\cot\), use:\( \tan \alpha = \frac{1}{\cot \alpha} \) and \( \tan \beta = \frac{1}{\cot \beta} \)

Step 4: Equate and Solve for Cotangent

Using the equations from the previous step:\[ x \cdot \frac{1}{\cot \alpha} = y \cdot \frac{1}{\cot \beta} \]Cross-multiply to solve for \(\cot \theta\):\[ x \cot \beta = y \cot \alpha \]

Step 5: Form the Cotangent Inclination Equation

Rearrange to find \(\cot \theta\):\[ y \cot \alpha - x \cot \beta = (y - x) \cot \theta \]This rearranges to:\[ \cot \theta = \frac{y \cot \alpha - x \cot \beta}{y - x} \]

Step 6: Identify the Correct Option

The correct answer matches option (A), which is \( \frac{y \cot \alpha - x \cot \beta}{y - x} \).

Key Concepts

Understanding Cotangent

In trigonometry, cotangent is one of the fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. Mathematically, cotangent is expressed as:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

This means that cotangent is the ratio of the adjacent side to the opposite side of a right-angled triangle.
Cotangent is particularly useful when dealing with slopes or inclinations, as it provides a direct measure of how steep a line is.

In the context of this exercise, we use cotangent to express the angles of elevation, \(\alpha\) and \(\beta\), from two different stations. The relationship allows us to compute the inclination angle \(\theta\) of the tower with respect to the horizontal, which is pivotal in real-world applications like construction or navigation.

Angles of Elevation

The angle of elevation is the angle between the horizontal line and the line of sight, when an observer looks upwards to an object. It is an essential concept in trigonometry and is used to determine heights or distances indirectly.

• For instance, angle \(\alpha\) is the angle of elevation from a station at distance \(x\) south of the tower's base.
• Similarly, \(\beta\) is the angle of elevation from another station at distance \(y\).

Understanding angles of elevation is essential for solving problems involving heights or elevations, such as finding the height of a building or mountain without physically measuring it.
By using the tangent function, where \(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\), we can relate the angle of elevation \(\alpha\) to the height of the tower. Solving for each station provides crucial data:

• From station \(x\), \(\tan(\alpha) = \frac{h}{x}\)
• From station \(y\), \(\tan(\beta) = \frac{h}{y}\)

These relationships help us connect the angle of elevation with the height of the tower and enable further calculations.

Inclination of the Tower

The inclination of a tower refers to its tilt from the vertical line. When a tower leans, its inclination needs to be quantified to address structural stability or precision in measurements.
In this problem, \(\theta\) represents the tower's angle of inclination to the horizontal.

This angle is crucial when determining a tower's exact position or orientation and handling tower maintenance. By using the angles of elevation \(\alpha\) and \(\beta\), along with their respective distances

• \(x\) and \(y\)

from the base, we can calculate \(\cot(\theta) = \frac{y \cot(\alpha) - x \cot(\beta)}{y - x} \).
This equation simplifies computations involving inclination by considering the differences in distances and elevation angles from multiple stations.
Ultimately, understanding and calculating the inclination ensures the safety and effectiveness of related engineering projects.