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 `alpha` then the distance between two consecutive poles, is

Options

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

Answer

To solve the problem, we need to find the distance between two consecutive poles given the conditions stated in the question. Let's break down the solution step by step.

### Step 1: Define the Variables
Let:
- \( h \) = height of the longest pole
- \( a \) = distance of the foot of the smallest pole from point \( O \)
- \( d \) = distance between two consecutive poles
- The total number of poles = 10

### Step 2: Determine the Position of the Poles
The distance of the smallest pole from point \( O \) is \( a \). The distances of the poles from \( O \) are:
- Smallest pole: \( a \)
- Second pole: \( a + d \)
- Third pole: \( a + 2d \)
- ...
- Longest pole (10th pole): \( a + 9d \)

### Step 3: Establish the Relationship Using Tangent
For the longest pole, the angle of elevation \( \alpha \) gives us:
\[ \tan(\alpha) = \frac{h}{a + 9d} \]

For the smallest pole, the angle of elevation is the same:
\[ \tan(\alpha) = \frac{h_1}{a} \]
where \( h_1 \) is the height of the smallest pole.

### Step 4: Set Up the Equations
From the above relationships, we can express \( h \) in terms of \( a \) and \( d \):
\[ h = (a + 9d) \tan(\alpha) \]
\[ h_1 = a \tan(\alpha) \]

### Step 5: Solve for \( d \)
We can express \( d \) in terms of \( h \), \( a \), and \( \tan(\alpha) \):
1. From the equation for the longest pole:
\[ h = (a + 9d) \tan(\alpha) \]
Rearranging gives:
\[ 9d = \frac{h}{\tan(\alpha)} - a \]
Thus,
\[ d = \frac{h - a \tan(\alpha)}{9 \tan(\alpha)} \]

### Step 6: Substitute \( \tan(\alpha) \)
Recall that \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \). Substituting this into our equation for \( d \):
\[ d = \frac{h - a \frac{\sin(\alpha)}{\cos(\alpha)}}{9 \frac{\sin(\alpha)}{\cos(\alpha)}} \]
This simplifies to:
\[ d = \frac{h \cos(\alpha) - a \sin(\alpha)}{9 \sin(\alpha)} \]

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