a) $$\sqrt {a + b} $$
b) $$\sqrt {ab} $$
c) $$\sqrt {a – b} $$
d) $$\sqrt {\frac{a}{b}} $$

Answer: b
Explanation: Let AB be the tower and P and Q are such points that PB = a, QB = b and angles of elevation at P and Q are 30° and 60° respectively

$$\eqalign{ & {\text{Let }}AB = h \cr & {\text{Now in right }}\Delta APB, \cr & \tan \theta = \frac{{{\text{Perpendicular}}}}{{{\text{Base}}}} = \frac{{AB}}{{PB}} \cr & \Rightarrow \tan {30^ \circ } = \frac{h}{a} \cr & \Rightarrow \frac{1}{{\sqrt 3 }} = \frac{h}{a}\,………..(i) \cr & {\text{Similarly in right }}\Delta AQB, \cr & \tan {60^ \circ } = \frac{{AB}}{{QB}} \cr & \Rightarrow \sqrt 3 = \frac{h}{b}\,………..(ii) \cr & {\text{Multiplying (i) and (ii)}} \cr & \frac{1}{{\sqrt 3 }} \times \sqrt 3 = \frac{h}{a} \times \frac{h}{b} \cr & 1 = \frac{{{h^2}}}{{ab}} \cr & {h^2} = ab \cr & h = \sqrt {ab} \cr & {\text{Height of the tower}} = \sqrt {ab} \cr} $$