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The angles of elevation of the top of a tower at two point, which are at distances a and b from the foot in the same horizontal line and on the same sides of the tower are complementary. The height of the tower is

(a) ab

(b) \(\sqrt{ab}\)

(c) \(\sqrt{\frac{a}b}\)

(d) \(\sqrt{\frac ba}\)

1 Answer

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Best answer

Correct option is (b) \(\sqrt{ab}\)

Let AB be the tower of height and C and D be the two given points of observation such that

BC = b, BD = a, ∠ACB = \(\beta\),

∠ADB = α.

In rt. \(\Delta\)ABC,

\(\tan \beta = \frac {AB}{b}\)

⇒ AB = b tan \(\beta\)    ...(i) 

In rt. \(\Delta\) ADB,

\(\tan\alpha= \frac {AB}{a}\)

⇒ AB = a tan α = a tan (90° – \(\beta\)

= a cot \(\beta\)    (\(\because\) α + \(\beta\) = 90°)    ...(ii)

Multiplying eqns (i) and (ii), we get

AB2 = b.tan \(\beta\) × a cot \(\beta\) 

⇒ AB2 = ab 

⇒ AB = \(\sqrt{ab}\)

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