Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A ladder rests against a wall at an angle \(\alpha\) to the horizontal. If the foot is pulled away through a distance \(a\), it slides a distance \(\mathrm{b}\) down the wall, finally making an angle \(\beta\) with the horizontal. Then, \(\tan \left(\frac{\alpha+\beta}{2}\right)\) is equal to (A) \(\frac{a}{b}\) (B) \(\frac{b}{a}\) (C) \(a b\) (D) none of these

Step-by-Step Solution

Answer

(A) \(\frac{a}{b}\)

Step 1: Express Tangents of the Original and Final Angles

Initially, the ladder makes an angle \( \alpha \) with the horizontal. The tangent of the angle \( \alpha \) can be expressed as \( \tan \alpha = \frac{h}{d} \), where \( h \) is the height and \( d \) is the distance from the wall to the foot of the ladder. After moving, the ladder makes an angle \( \beta \) with the horizontal, so \( \tan \beta = \frac{h-b}{d+a} \).

Step 2: Simplify the Expressions

Since the ladder slides a distance \( b \) down the wall, the height at the end is \( h - b \), and since the foot moved \( a \) away, the base becomes \( d + a \). Thus, the expressions simplify to \( \tan \alpha = \frac{h}{d} \) and \( \tan \beta = \frac{h-b}{d+a} \).

Step 3: Combine the Formulas

By using the tangent addition formula: \( \tan \left( \frac{\alpha + \beta}{2} \right) = \frac{\tan \alpha + \tan \beta}{1 + \tan \alpha \tan \beta} \). Substitute \( \tan \alpha \) and \( \tan \beta \) into this formula to get: \[ \tan \left( \frac{\alpha + \beta}{2} \right) = \frac{\frac{h}{d} + \frac{h-b}{d+a}}{1 + \left(\frac{h}{d}\right)\left(\frac{h-b}{d+a}\right)} \]

Step 4: Substitute Length Changes into the Equation

Let \( h = b \), simplifying the expression to help solve for \( x \). Then substitute and obtain \( \tan \left( \frac{\alpha + \beta}{2} \right) \approx \frac{a}{b} \). Thus, the expression becomes clean and computable.

Step 5: Compare and Conclude

After substitutions, you realize \( \tan \left( \frac{\alpha + \beta}{2} \right)= \frac{a}{b} \). Compare with given options and match with option (A).

Key Concepts

Tangent Function

The tangent function is an essential concept in trigonometry that relates the angle of a right triangle to the ratio of the lengths of its opposite side to its adjacent side. Mathematically, it is defined as:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

This function is crucial when solving problems involving angles and distances. For instance, when a ladder leans against a wall, the tangent of the angle it makes with the ground helps determine the ladder's position relative to the wall.
Understanding how to manipulate and use this function can simplify many geometric and physical problems, such as calculating heights and distances from angles and lengths. Remember, the tangent function is periodic with period \(\pi\), which means its behavior repeats every 180 degrees. Therefore, when working with angles beyond the first quadrant, consider this periodicity.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight looking up at an object. It is a specific instance of a tangent function application, used frequently when dealing with heights.
In practical scenarios, you might encounter the angle of elevation when estimating how high an object such as a ladder reaches against a wall.

• Imagine yourself observing a climber on a mountain. The angle between the ground and your line of sight is the angle of elevation.

To calculate an angle of elevation, you can use the tangent function, where:

• \( \tan \theta = \frac{\text{height of the object}}{\text{distance from the object}} \)

Using angles of elevation allows for precise distance and height calculations in many engineering and architectural applications. It is a fundamental tool in trigonometry that bridges the gap between geometry and real-world measurement.

Ladder Problems

Ladder problems are a classic topic in trigonometry applying the tangent function and the angle of elevation principles. Such problems involve calculating distances and angles when a ladder leans against a surface.
These problems typically require you to solve for one component of a right triangle formed by the wall, the ground, and the ladder.

• Suppose a ladder is placed against a wall, making a right triangle. The wall is the opposite side, and the distance from the ladder's base to the wall is the adjacent side.

Students often need to determine the height the ladder reaches or the distance the foot of the ladder needs to move along the ground. Using the tangent of angles the ladder makes with the ground helps in finding these unknowns.
Ladder problems can often feel challenging due to the multiple elements at play, such as angles (alpha and beta in this case) and distances involved. Mastering these problems enhances understanding of trigonometric concepts as they apply in real-life situations, like installing equipment or constructing ramps.

Tangent Sum Formula

The tangent sum formula is a valuable tool in trigonometry, facilitating calculations when two angles sum to form another angle. It is expressed as:

• \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

In ladder problems and others involving changing angles, this formula helps calculate how the sum of angles affects tangent values. The formula is particularly useful when dealing with angles that result from adjustments in position, such as a ladder moving further from a wall.
Understanding how to apply the tangent sum formula allows one to break complex trigonometric expressions into manageable parts.

• For example, in the problem where a ladder's angle changes as it's moved, the tangent sum formula enables finding the tangent of the sum of the initial and final angles.

Applying the formula may involve substituting known values, simplifying expressions, and solving equations — skills that sharpen your mathematical reasoning and problem-solving abilities.