The angle of elevation of a bird from a point 12 metres above the lake is 30⁰ and the angle of depression of its reflection in the lake is 60⁰. Find the distance of the bird from the point of observation.

Dear Student,

Please find below the solution to the asked query:

From given information we form our diagram , As :



Here , Level of lake  =  DE , Bird at point " C "  and reflection at " F " and observer at point " A " .

Let , Distance from point of observation to bird =  AC =  x

And

BC  =  h m  , AD  =  BE  =  12 m  , EF  =  12 +  h  (  As reflection have same distance from surface )

We know ,  $\tan \mathrm{\theta } = \frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$ , So

In triangle ABF 

$$\begin{gathered} \tan 60° = \frac{\mathrm{FB}}{\mathrm{AB}} \\ \Rightarrow \sqrt{3} = \frac{12 + \mathrm{h}}{\mathrm{AB}} ( \mathrm{We} \mathrm{know} \tan 60° =\sqrt{3} ) \\ \mathbf{\Rightarrow }\mathbf{AB}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{12}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{h}}{\sqrt{\mathbf{3}}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{)}\mathbf{ }\mathbf{ } \end{gathered}$$

And

In triangle ABC

$$\begin{gathered} \tan 30° = \frac{\mathrm{BC}}{\mathrm{AB}} \\ \Rightarrow \frac{1}{\sqrt{3}} = \frac{\mathrm{h}}{\mathrm{AB}} ( \mathrm{We} \mathrm{know} \tan 30° =\frac{1}{\sqrt{3}} ) \\ \mathbf{\Rightarrow }\mathbf{AB}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{h}\sqrt{\mathbf{3}}\mathbf{ } \\ \mathrm{So}, \mathrm{From} \mathrm{equation} 1 \mathrm{we} \mathrm{get} \\ \mathrm{h}\sqrt{3} = \frac{12 + \mathrm{h}}{\sqrt{3}} \\ 3 \mathrm{h} = 12 + \mathrm{h} \\ 2 \mathrm{h} = 12 \\ \mathbf{h}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{6} \end{gathered}$$
We know $\mathrm{Sin} \mathrm{\theta } = \frac{\mathrm{Opposite}}{\mathrm{Hypotenuse}}$
So,
In triangle ABC

$$\begin{gathered} \mathrm{Sin} 30° = \frac{\mathrm{BC}}{\mathrm{AC}} \\ \Rightarrow \frac{1}{2} = \frac{6}{x} ( \mathrm{We} \mathrm{know} \mathrm{Sin} 30° =\frac{1}{2}) \\ \mathbf{\Rightarrow }\mathit{x}\mathbf{=}\mathbf{ }\mathbf{12} \end{gathered}$$
Therefore,

Distance from point of observation to bird = 12 m                                                ( Ans )

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