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

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, \(\tan \left(\frac{\alpha+\beta}{2}\right)\) 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: Understand the Problem

A ladder initially makes an angle \(\alpha\) with the horizontal. When the foot of the ladder is moved by distance \(a\), it causes the top end of the ladder to slide down the wall by a distance \(b\), changing the angle to \(\beta\). We need to determine \(\tan \left(\frac{\alpha+\beta}{2}\right)\).

Step 2: Recognize the Triangle Relationships

The ladder, floor, and wall form two right-angled triangles before and after the movement. Initially, the height from the wall is \(c \sin \alpha\) and the base is \(c \cos \alpha\), where \(c\) is the ladder's length. After movement, the height is \(c \sin \beta\) and the base is \(c \cos \beta\).

Step 3: Establish Equations from Movement

From the initial and final positions, use the displacement information: the foot of the ladder moves by \(a\), and the top slides down the wall by \(b\). Thus, \(c \cos \beta = c \cos \alpha + a\) and \(c \sin \beta = c \sin \alpha - b\).

Step 4: Use Half-Angle Formulas

We apply the tangent half-angle formula: \(\tan \left(\frac{\alpha + \beta}{2}\right) = \sqrt{\frac{1 - \cos(\alpha + \beta)}{1 + \cos(\alpha + \beta)}}\). Since \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\), we substitute using our expressions developed in previous steps.

Step 5: Simplify Expressions

Substitute \(\cos(\alpha)\), \(\cos(\beta)\), \(\sin(\alpha)\), and \(\sin(\beta)\) in the tangent formula. After simplifying, it becomes evident that \(\tan \left(\frac{\alpha+\beta}{2}\right) = \frac{a}{b}\).

Step 6: Confirm Answer Choice

By the calculations, the correct answer corresponds to option (A), where \(\tan \left(\frac{\alpha+\beta}{2}\right) = \frac{a}{b}\).

Key Concepts

Angle of a Ladder

When a ladder leans against a wall, the angle it makes with the ground is crucial for understanding its stability. This angle, often denoted as \( \alpha \), is known as the angle of a ladder. Adjusting this angle can impact both the height the ladder reaches on the wall and its base distance from the wall.

• The larger the angle, closer to 90 degrees, the higher the ladder reaches up the wall.
• Conversely, a smaller angle, approaching 0 degrees, means the ladder is more horizontal and reaches less height.

As the ladder moves (for instance, when the foot is pulled away from the wall), this angle changes, influencing the position and distance the ladder covers. Understanding this concept is pivotal in problems related to ladders leaning against walls, where the angle change is often linked to the distances moved by the ladder's ends.

Tangent Half-Angle Formula

The tangent half-angle formula is a trigonometric identity that helps in determining the tangent of half the sum of two angles. This is particularly useful in problems that involve angle changes, such as when a ladder's position is adjusted.The formula is given as:\[ \tan \left( \frac{\alpha + \beta}{2} \right) = \sqrt{\frac{1 - \cos(\alpha + \beta)}{1 + \cos(\alpha + \beta)}} \]

Application in Ladder Problems:

To use this formula, one needs to first determine \( \cos(\alpha + \beta) \). This involves knowing the individual cosines and sines of the angles \( \alpha \) and \( \beta \). In the case of a leaning ladder, these trigonometric functions are connected to the distances moved:

• The initial cosine and sine come from the triangle formed by the ladder's initial position.
• After movement, the new positions provide updated cosine and sine values, which are used in the tangent half-angle formula.

Right Triangles

Right triangles are fundamental in understanding how a ladder interacts with a wall since the ladder, the wall, and the ground form a right triangle. In this setup, the right angle is between the wall and the ground. Right triangles have properties that simplify calculations, especially involving angles and distances. Important points include:

• The Pythagorean theorem can be applied to find relationships between the ladder length, height on the wall, and distance from the wall.
• Trigonometric ratios like sine, cosine, and tangent relate these side lengths and the angles between them.

Knowing these properties helps in setting up equations that define how the movement of the ladder's foot affects the top's sliding distance along the wall. With each position change, the new triangle's sides can be calculated, leading to insights about the new angle and distances involved.