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 = \(\sqrt{ab}\)