$n$ poles standing at equal distances on a
straight road subtend the same angle $α$ at a point $O$ on the
road. If the height of the largest pole is $h$ and the distance
of the foot of the smallest pole from $O$ is $a$ , the distance
between two consecutive poles is

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$\frac{h \sin α - a \cos α}{(n - 1) \sin α}$

$\frac{h \cos α - a \cos α}{(n - 1) \cos α}$

$\frac{h \cos α - a \sin α}{(n - 1) \sin α}$

$\frac{h \sin α - a \cos α}{(n - 1) \cos α}$

answer is C.

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Detailed Solution

Let $A_{1}, A_{2} \dots A_{n}$ be the feet of the $n$ poles subtending
angle $α$ at $O$ such that $O A_{1} = a,$ if $d$ be the distance
between two consecutive poles

$\begin{array}{l} O A_{2} = a + d \\ O A_{3} = a + 2 d \\ ⋮ \\ O A_{n} = a + ∣ (n - 1) d \end{array}$

Since the height of the pole at $A_{n}$ is $h$

$\begin{array}{l} O A_{n} = h \cot α . \\ \Rightarrow a + (n - 1) d = h \cot α \\ \Rightarrow d = \frac{h \cot α - a}{n - 1} = \frac{h \cos α - a \sin α}{(n - 1) \sin α} \end{array}$