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 the foot of the smallest pole from $$O$$ is $$a$$; then the distance between two consecutive poles, is

We have 10 vertical poles standing at equal distances on a straight line, all subtending the same angle of elevation $$\alpha$$ at a point $$O$$ on the same line and same side. The height of the longest pole is $$h$$, and the distance from $$O$$ to the foot of the smallest pole is $$a$$. We need to find the distance between two consecutive poles.

Since there are 10 poles, there are 9 intervals between them. Let the distance between two consecutive poles be $$d$$. Therefore, the total distance from the foot of the smallest pole to the foot of the longest pole is $$9d$$.

The smallest pole is closest to $$O$$, so its distance from $$O$$ is $$a$$. The longest pole is farthest from $$O$$, so its distance from $$O$$ is $$a + 9d$$.

Each pole subtends the same angle $$\alpha$$ at $$O$$. For any pole, if its distance from $$O$$ is $$x$$ and its height is $$y$$, then $$\tan \alpha = \frac{y}{x}$$, so $$y = x \tan \alpha$$.

Therefore, the height of the smallest pole (at distance $$a$$) is $$a \tan \alpha$$. The height of the longest pole (at distance $$a + 9d$$) is $$h$$, so we have:

$$h = (a + 9d) \tan \alpha$$

We know that $$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$$, so substitute:

$$h = (a + 9d) \cdot \frac{\sin \alpha}{\cos \alpha}$$

Multiply both sides by $$\cos \alpha$$:

$$h \cos \alpha = (a + 9d) \sin \alpha$$

Expand the right side:

$$h \cos \alpha = a \sin \alpha + 9d \sin \alpha$$

Isolate the term containing $$d$$:

$$h \cos \alpha - a \sin \alpha = 9d \sin \alpha$$

Solve for $$d$$:

$$d = \frac{h \cos \alpha - a \sin \alpha}{9 \sin \alpha}$$

Comparing with the options, this matches Option B.

Hence, the correct answer is Option B.