The angle of elevation of a cloud from a point h metres above the surface of a lake is θ and the angle of depression of its reflection in the lake is φ Prove that the height of the cloud above the lake is h [(tan φ + tan θ)/ (tan φ - tan θ)].

Solution:

Consider P as the cloud and Q as its reflection in the lake.

Consider A as the point of observation where AB = h.

Consider the height of the cloud above the lake as x.

AL = d.

In ∆PAL,

(x - h)/d = tan θ …. (1)

In ∆QAL,

(x + h)/d = tan φ …. (2)

Dividing equation (2) by (1),

(x + h)/ (x - h) = tan φ/ tan θ

It can be written as

2x/2h = (tan φ + tan θ)/ (tan φ - tan θ)

So we get

x/h = (tan φ + tan θ)/ (tan φ - tan θ)

By cross multiplication

x = h [(tan φ + tan θ)/ (tan φ - tan θ)]

Therefore, it is proved.

 ✦ Try This: If the angle of elevation of a cloud from a point h metres above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is 2 h sec α/ (tan β - tan α).

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.4 Sample Problem 3

The angle of elevation of a cloud from a point h metres above the surface of a lake is θ and the angle of depression of its reflection in the lake is φ

Summary:

The angle of elevation of a cloud from a point h metres above the surface of a lake is θ and the angle of depression of its reflection in the lake is φ. It is proved that the height of the cloud above the lake is h [(tan φ + tan θ)/ (tan φ - tan θ)]

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