A man standing on the line joining the two poles finds that the top of the poles make an angle of elevation of $$60^\circ$$ and $$45^\circ$$ respectively. After walking for sometime towards the other pole, the angles change to $$30^\circ$$ and $$60^\circ$$ respectively. The ratio of the height of the poles is :

Let 'a' and 'b' be the heights of the two poles

X be the initial position of the man and the angles of elevation be $$60^0$$ and $$45^0$$

Distance between pole 1 and X  = $$\ \frac{\ a}{\sqrt{\ 3}}$$ and distance between pole 2 and X = b

Let Y be the position of the man after walking sometime towards the other pole = 

Distance between Y and pole 1 = $$\ \ a\sqrt{\ 3}$$ and distance between the pole 2 and Y = $$\ \frac{\ b}{\sqrt{\ 3}}$$

Since the distance between the poles will remain the same 

$$\ \frac{\ a}{\sqrt{\ 3}}$$ + b = $$\ \ a\sqrt{\ 3}$$ + $$\ \frac{\ b}{\sqrt{\ 3}}$$

$$\ \frac{\ a}{b}=\ \ \frac{\ \sqrt{\ 3}-1}{2}$$
A is the correct answer.