Let 10 vertical poles standing at equal distances on a straight line , subtend the same angle of elevation `alpha` at a point O on this line and all the poles are on the same side of O. If the height of the longest pole is h and the distance of foot the smallest pole form O is a then the distance between two consecutive poles, is

Options

A. `( h sin alpha + a cos alpha)/(9 cos alpha)`
B. `(h cos alpha - a sin alpha)/(9 sin alpha)`
C. `( h sin alpha + a cos alpha)/(9 sin alpha)`
D. `(h cos alpha - a sin alpha)/(9 cos alpha)`

Answer

To solve the problem, we need to find the distance between two consecutive poles given that they subtend the same angle of elevation \( \alpha \) at a point \( O \) on the line of the poles. Here’s how to approach the problem step by step:

### Step 1: Understand the setup
We have 10 vertical poles standing at equal distances on a straight line. The height of the longest pole is \( h \), and the distance from the point \( O \) to the foot of the smallest pole is \( a \). The distance between two consecutive poles is denoted as \( D \).

### Step 2: Define the distances
Since there are 10 poles, and they are equidistant, the distances from the point \( O \) to the foot of each pole can be expressed as follows:
- Distance to the first pole (smallest): \( a \)
- Distance to the second pole: \( a + D \)
- Distance to the third pole: \( a + 2D \)
- ...
- Distance to the tenth pole (longest): \( a + 9D \)

### Step 3: Apply the angle of elevation
The angle of elevation \( \alpha \) is the same for all poles. Therefore, we can use the tangent of the angle of elevation to relate the height of the poles to their distances from point \( O \).

For the longest pole (height \( h \)):
\[ \tan(\alpha) = \frac{h}{a + 9D} \]

For the smallest pole (height \( 0 \)):
\[ \tan(\alpha) = \frac{0}{a} \quad \text{(this is not useful since it equals zero)} \]

### Step 4: Set up the equation
From the equation for the longest pole, we can express it as:
\[ h = (a + 9D) \tan(\alpha) \]

### Step 5: Solve for \( D \)
Rearranging the equation gives:
\[ h = a \tan(\alpha) + 9D \tan(\alpha) \]
\[ h - a \tan(\alpha) = 9D \tan(\alpha) \]
\[ D = \frac{h - a \tan(\alpha)}{9 \tan(\alpha)} \]

### Step 6: Substitute \( \tan(\alpha) \)
Using the identity \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \), we can substitute:
\[ D = \frac{h - a \frac{\sin(\alpha)}{\cos(\alpha)}}{9 \frac{\sin(\alpha)}{\cos(\alpha)}} \]
\[ D = \frac{(h \cos(\alpha) - a \sin(\alpha))}{9 \sin(\alpha)} \]

### Final Answer
Thus, the distance between two consecutive poles is:
\[ D = \frac{h \cos(\alpha) - a \sin(\alpha)}{9 \sin(\alpha)} \]