The angles of elevation of a cliff at a point A on the ground and at a point B 100 mt vertically above A are `alpha` are `beta` respectively.The height of the cliff in metres is

Options

A. `(100 cot alpha )/(cot alpha - cot beta)`
B. `(100 cot beta)/(cot alpha - cot beta)`
C. `(100 cot beta)/(cot beta-cot alpha )`
D. `(100 cot beta)/(cot beta+ cot alpha )`

Answer

To solve the problem, we need to find the height of the cliff using the angles of elevation from two different points. Let's break it down step by step.

### Step 1: Understand the problem
We have two points:
- Point A is on the ground.
- Point B is 100 meters above Point A.

From Point A, the angle of elevation to the top of the cliff is α, and from Point B, the angle of elevation is β.

### Step 2: Set up the relationships using trigonometry
Using the tangent function, we can set up the following equations based on the angles of elevation:

1. From Point A:
\[ \tan(\alpha) = \frac{h}{d} \]
where \( h \) is the height of the cliff and \( d \) is the horizontal distance from Point A to the base of the cliff.

2. From Point B (which is 100 meters above Point A):
\[ \tan(\beta) = \frac{h - 100}{d} \]

### Step 3: Rearranging the equations
From the first equation, we can express \( h \) in terms of \( d \):
\[ h = d \cdot \tan(\alpha) \]

From the second equation, we can express \( h \) as well:
\[ h - 100 = d \cdot \tan(\beta) \implies h = d \cdot \tan(\beta) + 100 \]

### Step 4: Set the two expressions for \( h \) equal to each other
Now we have two expressions for \( h \):
\[ d \cdot \tan(\alpha) = d \cdot \tan(\beta) + 100 \]

### Step 5: Isolate \( d \)
Rearranging gives us:
\[ d \cdot \tan(\alpha) - d \cdot \tan(\beta) = 100 \]
\[ d (\tan(\alpha) - \tan(\beta)) = 100 \]
\[ d = \frac{100}{\tan(\alpha) - \tan(\beta)} \]

### Step 6: Substitute \( d \) back to find \( h \)
Now that we have \( d \), we can substitute it back into one of the equations for \( h \):
\[ h = d \cdot \tan(\alpha) = \frac{100}{\tan(\alpha) - \tan(\beta)} \cdot \tan(\alpha) \]
\[ h = \frac{100 \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]

### Step 7: Convert to cotangent form
Using the identity \( \tan(x) = \frac{1}{\cot(x)} \):
\[ h = \frac{100 \cdot \frac{1}{\cot(\alpha)}}{\frac{1}{\cot(\alpha)} - \frac{1}{\cot(\beta)}} \]
This simplifies to:
\[ h = \frac{100 \cdot \cot(\alpha) \cdot \cot(\beta)}{\cot(\beta) - \cot(\alpha)} \]

### Final Result
Thus, the height of the cliff \( h \) is given by:
\[ h = \frac{100 \cdot \cot(\beta)}{\cot(\beta) - \cot(\alpha)} \]