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The angles of elevation of the top of a tower standing on a horizontal plane, from two points on a line passing through its foot at distances $a$ and $b,$ respectively, are complementary angles. If the line joining the two points subtends an angle $θ$ at the top of the tower, then if $a > b, \sin θ =$

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$\frac{2 \sqrt{a b}}{a - b}$

$\frac{a - b}{a + b}$

$\frac{2 \sqrt{a b}}{a + b}$

$\frac{a + b}{a - b}$

answer is A.

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Detailed Solution

$\tan α = h / a, \cot α = h / b .$

$\Rightarrow h = \sqrt{a b} .$

$\begin{array}{l} \tan (α + θ) = a / h \\ \Rightarrow \frac{\tan α + \tan θ}{1 - \tan α \tan θ} = \frac{a}{h} . \end{array}$

$\Rightarrow h \tan α + h \tan θ = a - a \tan α \tan θ$

$\begin{array}{l} \Rightarrow \tan θ = \frac{a - b}{h + a \tan α} = \frac{a - b}{2 \sqrt{a b}} . \\ \Rightarrow \sin θ = \frac{a - b}{a + b} \end{array}$

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