From an aeroplane vertically over a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be `alpha` and `beta`. The height of the aeroplane above the road is

Options

A. `(tan alpha + tan beta)/(tan alpha tan beta)`
B. `(tan alpha tan beta)/(tan alpha + tan beta)`
C. `(cot alpha cot beta)/(cot alpha + cot beta)`
D. none of these

Answer

To find the height of the aeroplane above the road given the angles of depression \( \alpha \) and \( \beta \) to two consecutive milestones, we can follow these steps:

### Step 1: Understand the Geometry
Consider the situation where the aeroplane is at a height \( h \) above the road. Let the distance from the point directly below the aeroplane to the first milestone (where angle of depression is \( \alpha \)) be \( 1 - x \) and to the second milestone (where angle of depression is \( \beta \)) be \( x \).

### Step 2: Apply the Tangent Function
From the right triangle formed by the height of the aeroplane and the distance to the milestones, we can use the tangent function:
- For the first milestone:
\[ \tan(\alpha) = \frac{h}{1 - x} \]
Rearranging gives:
\[ h = (1 - x) \tan(\alpha) \]

- For the second milestone:
\[ \tan(\beta) = \frac{h}{x} \]
Rearranging gives:
\[ h = x \tan(\beta) \]

### Step 3: Set the Two Expressions for \( h \) Equal
Since both expressions equal \( h \), we can set them equal to each other:
\[ (1 - x) \tan(\alpha) = x \tan(\beta) \]

### Step 4: Solve for \( x \)
Expanding and rearranging the equation:
\[ \tan(\alpha) - x \tan(\alpha) = x \tan(\beta) \]
\[ \tan(\alpha) = x (\tan(\alpha) + \tan(\beta)) \]
\[ x = \frac{\tan(\alpha)}{\tan(\alpha) + \tan(\beta)} \]

### Step 5: Substitute \( x \) Back to Find \( h \)
Now substitute \( x \) back into either expression for \( h \). Using \( h = x \tan(\beta) \):
\[ h = \left(\frac{\tan(\alpha)}{\tan(\alpha) + \tan(\beta)}\right) \tan(\beta) \]
\[ h = \frac{\tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

### Final Result
Thus, the height of the aeroplane above the road is given by:
\[ h = \frac{\tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

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