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Three poles of height $a, b, c$ stand on the same side of a road and subtend an angle of $45 °$ at a point on the line joining their feet. The pole of height a subtends an angle $α$ at the foot of the pole of height $b$ which subtends an angle $α$ at the foot of the pole with height $c,$ if $a > b > c,$ then $c o t α - c o t β =$

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$\frac{a c - b^{2}}{a b}$

$\frac{b c - a^{2}}{a b}$

$\frac{a c - b^{2}}{b c}$

$\frac{a b - c^{2}}{b c}$

answer is A.

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Detailed Solution

$\begin{aligned} P Q = a - b \Rightarrow Q L = a - b \\ \Rightarrow O A = a - b \\ \Rightarrow a \cot α = a - b \\ \Rightarrow \cot α = \frac{a - b}{a} \end{aligned}$

Similarly $\cot β = \frac{b - c}{b} .$

So that $\cot α - \cot β = \frac{c}{b} - \frac{b}{a} = \frac{a c - b^{2}}{a b}$

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