A ladder rests against a wall making an angle `alpha` with the horizontal. The foot of the ladder is pulled away from the wall through a distance ‘a’ so that it slides a distance ‘b’ down the wall making an angle beta with the horizontal then the value of `(a)/(b)` is

Options

A. `tan""(alpha+beta)/(2)`
B. `tan""(2)/(alpha+beta)`
C. `alpha+beta`
D. `alpha+beta`

Answer

To solve the problem, we need to find the ratio \( \frac{a}{b} \) when a ladder rests against a wall and slides down as its foot is pulled away from the wall. Let's break down the solution step by step.

### Step 1: Understand the Geometry
We have a ladder of length \( L \) resting against a wall, making an angle \( \alpha \) with the horizontal. When the foot of the ladder is pulled away from the wall by a distance \( a \), it slides down the wall by a distance \( b \), making a new angle \( \beta \) with the horizontal.

### Step 2: Identify the Components
1. **Initial Position**:
- The height of the ladder on the wall can be expressed as \( h_1 = L \sin \alpha \).
- The distance from the wall to the foot of the ladder is \( d_1 = L \cos \alpha \).

2. **Final Position**:
- After sliding down, the height of the ladder on the wall becomes \( h_2 = L \sin \beta \).
- The distance from the wall to the foot of the ladder is \( d_2 = L \cos \beta \).

### Step 3: Calculate the Distances
- The horizontal distance the foot of the ladder has been pulled away is:
\[ a = d_2 - d_1 = L \cos \beta - L \cos \alpha = L (\cos \beta - \cos \alpha) \]

- The vertical distance the ladder has slid down is:
\[ b = h_1 - h_2 = L \sin \alpha - L \sin \beta = L (\sin \alpha - \sin \beta) \]

### Step 4: Formulate the Ratio
Now, we can find the ratio \( \frac{a}{b} \):
\[ \frac{a}{b} = \frac{L (\cos \beta - \cos \alpha)}{L (\sin \alpha - \sin \beta)} \]
Since \( L \) is common in both the numerator and the denominator, it cancels out:
\[ \frac{a}{b} = \frac{\cos \beta - \cos \alpha}{\sin \alpha - \sin \beta} \]

### Step 5: Apply Trigonometric Identities
Using the trigonometric identities:
- The difference of cosines can be expressed as:
\[ \cos \beta - \cos \alpha = -2 \sin\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right) \]

- The difference of sines can be expressed as:
\[ \sin \alpha - \sin \beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right) \]

### Step 6: Substitute and Simplify
Substituting these identities into our ratio:
\[ \frac{a}{b} = \frac{-2 \sin\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right)}{2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)} \]
This simplifies to:
\[ \frac{a}{b} = -\frac{\sin\left(\frac{\beta + \alpha}{2}\right) \sin\left(\frac{\beta - \alpha}{2}\right)}{\cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)} \]

### Conclusion
Thus, the final expression for the ratio \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \tan\left(\frac{\alpha + \beta}{2}\right) \]