The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

Question: The angle of elevation of the top of a tower of height x metre from a point on the ground is found to be 60°. By going y metre away from that point, it becomes 30°. Which one of the following relations is correct?

Solution:
Given that,
The angle of elevation of the top of a tower of height x meter from a point on the ground is found to be 60°.
By going y metre away from that point, it becomes 30°.

According to the question,
tan60° = AB/BC
⇒ √3 = x/BC
⇒ BC = x/√3 .........(1)

Again,
tan30° = AB/BD
⇒ 1/√3 = AB/BD
⇒ 1/√3 = x/BD
⇒ BD = √3x .........(2)

Now,
BD = BC + CD
⇒ √3x = (x/√3) + y [From equation (1 and 2)]
⇒ √3x = (x + √3y)/√3
⇒ 3x = x + √3y
⇒ 3x - x = √3y
⇒ 2x = √3y