Exactly midway between the foot of two towers P and Q, the angles of elevation of their tops are 45^o and 60^o, respectively. The ratio of the heights of P and Q is :

$ 1 : \sqrt{3}$

⇒ Let the height of towers P and Q be CE and AB and D is the mid point on EB.

ED = DB

⇒ In triangle CED

⇒ tan \({45}^\circ\) = \(\frac{CE}{ED}\)

⇒ 1 = \(\frac{CE}{ED}\)

⇒ ED = CE

In triangle ADB

⇒ tan \({60}^\circ\) = \(\frac{AB}{DB}\)

⇒ \(\sqrt {3 }\) = \(\frac{AB}{DB}\)

⇒ AB = \(\sqrt {3 }\)DB or  \(\sqrt {3 }\)ED

Ratio of height of tower P & Q = CE : AB = ED : \(\sqrt {3 }\)ED

⇒ 1 : \(\sqrt {3 }\)

Therefore, ratio of heights of p and Q is 1 : \(\sqrt {3 }\)