A ladder rests against a wall at an angle $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance p, so that it slides a distance q down the wall making an angle $\beta$ with the horizontal, then $\frac{p}{q}$ is equal to :

Question

Accepted Answer

To solve this problem, we need to analyze the situation described and understand the relationship between the distances moved by the ladder and the angles involved.

1. **Understanding the Problem:**
   - A ladder is initially resting against a wall, making an angle $\alpha$ with the horizontal.
   - The foot of the ladder is pulled away from the wall by a distance $p$, causing the ladder to slide down the wall by a distance $q$.
   - After moving, the ladder makes a new angle $\beta$ with the horizontal.

2. **Geometric Interpretation:**
   - Initially, the ladder forms a right triangle with the wall and the ground, where the ladder is the hypotenuse.
   - When the foot of the ladder is moved by $p$, the top of the ladder slides down by $q$.
   - The new position of the ladder also forms a right triangle with the wall and the ground.

3. **Trigonometric Relationships:**
   - Initially, the horizontal distance (base of the triangle) is $L \cos(\alpha)$, and the vertical distance (height of the triangle) is $L \sin(\alpha)$, where $L$ is the length of the ladder.
   - After moving, the new horizontal distance is $L \cos(\beta)$, and the new vertical distance is $L \sin(\beta)$.

4. **Relating $p$ and $q$:**
   - The change in horizontal distance is $p = L \cos(\beta) - L \cos(\alpha)$.
   - The change in vertical distance is $q = L \sin(\alpha) - L \sin(\beta)$.

5. **Using Trigonometric Identities:**
   - We need to find the ratio $\frac{p}{q}$.
   - Using the sum-to-product identities, we have:
     $$
     \cos(\beta) - \cos(\alpha) = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)
     $$
     $$
     \sin(\alpha) - \sin(\beta) = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)
     $$
   - Therefore, the ratio becomes:
     $$
     \frac{p}{q} = \frac{-2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)}{2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right)}
     $$
   - Simplifying this, we get:
     $$
     \frac{p}{q} = -\tan\left(\frac{\alpha + \beta}{2}\right)
     $$

6. **Conclusion:**
   - The negative sign indicates the direction of movement, but the magnitude of the ratio is $\tan\left(\frac{\alpha + \beta}{2}\right)$.
   - Therefore, the correct answer is Option 2: $\tan\left(\frac{\alpha + \beta}{2}\right)$.

7. **Why Other Options Are Incorrect:**
   - **Option 1 ($\tan(\alpha+\beta)$):** This option does not correctly represent the relationship between $p$ and $q$ as derived from the trigonometric identities.
   - **Option 3 ($\cot(\alpha+\beta)$):** This is the reciprocal of Option 1 and similarly does not match the derived relationship.
   - **Option 4 ($\cot(\frac{\alpha+\beta}{2})$):** This is the reciprocal of the correct answer and does not represent the correct ratio.

By following the trigonometric identities and understanding the geometric changes, we confirm that Option 2 is indeed the correct answer.