Multiple choice

From the top of a lighthouse, the angles of depression of two stations on opposite sides of it at distance $a$ apart are $\displaystyle \alpha $ and $\displaystyle \beta $. The height of the lighthouse is

  1. $\displaystyle \frac{a}{\cot \alpha \cot \beta }$
  2. $\displaystyle \frac{a}{\cot \alpha +\cot \beta }$
  3. $\displaystyle \frac{a\cot \alpha \cot\beta }{\cot \alpha +\cot \beta }$
  4. $\displaystyle \frac{a\tan \alpha \tan \beta }{\cot \alpha +\cot \beta }$
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B Correct answer
Explanation

Let h be the height. The two distances from the base are h * cot(alpha) and h * cot(beta). Since they are on opposite sides, h * cot(alpha) + h * cot(beta) = a. Thus, h = a / (cot(alpha) + cot(beta)).

AI explanation

Let h be the height of the lighthouse and x and y be the horizontal distances from the base to the two stations. The total distance between the stations is a, so x plus y equals a. Using the trigonometric ratios for the angles of depression, cot alpha equals x over h and cot beta equals y over h. Adding these two equations gives cot alpha plus cot beta equals (x plus y) over h, which simplifies to cot alpha plus cot beta equals a over h. Solving this equation for h gives the height of the lighthouse as a divided by (cot alpha plus cot beta).

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