Question

The angle of elevation of a bird from a point 'h' metres above a lake is angle alpha and the angle of depression of its reflection in the lake is angle beta. Find the height of the bird.

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Answer 1

Let a be a point h meters over the lake AF and B be the situation of the Bird. 
Attract a line parallel to EF from A on BD at C. 
But, BF = DF
Let, BC = m
so, BF = (m + h)
⇒ BF = DF = (m + h) metres
Consider ΔBAC,
AB = m cosec α ---------- (1)
and, AC = m cot α
Consider ΔACD,AC = (2h + m) cot β
Therefore, m cot α = (2h + m) cot β
⇒ m = 2h cot β / (cot α -  cot β)
Substituting the value of m in (1) we get,
AB = cosec α [2h cot β / (cot α -  cot β)] = 2h sec α / (tan β - tan α)

Now, 2h sec α / (tan β - tan α) = 2h AB/AC / (FB + h/AC - FB - h/AC)

= 2h*AB / (FB + h - FB + h)

= 2h*AB/2h

= AB is the Distance

Hope It Helps

And Fig Is Attached

Answer 2

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HERE IS YOUR ANSWER ☞
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let's the hight = X m

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GIVEN

=> EF = h meter

=> angle(AED) = alpha

and

=> angle(DEB) = beta

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Now

=> AC = BC = X ____(low of reflection)

=> CD = EF = h
_____________

than

=> AD = AC - CD = (X - h)

and

=> BD = BC + CD = (X + h)
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In △DEB,

$= > tan \beta \: = \frac{BD}{DE } \\ \\ = > tan \beta \: = \frac{x + h}{DE} \\ \\ = > DE = \frac{x + h}{tan \beta } \: \: \: ........eq _{1}$

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In △AED

$= > tan \alpha = \frac{AD}{DE} = \frac{x - h}{DE} \\ \\ = > DE \: = \frac{x - h}{tan \alpha } \: \: \: \: ......eq _{2}$

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$from \: eq _{1} \: and \: eq _{2} \\ \\ = > \frac{x + h}{tan \beta } = \frac{x - h}{tan \alpha } \\ \\ = > tan \alpha (x + h) = tan \beta (x - h) \\ \\ = > xtan \alpha \: + \: h.tan \alpha = xtan \beta - h.tan \beta \\ \\ = > x(tan \alpha - tan \beta ) = - h(tan \beta + tan \alpha ) \\ \\ = > x = - h(\frac{tan \beta + tan \alpha }{tan \alpha - tan \beta } ) \\ \\ = > x = h( \frac{tan \beta + tan \alpha }{tan \beta - tan \alpha } ) \: \: \: \: \: \: ......answer$

_________________

HENCE,

$the \: hight \: of \: the \: bird \: = \: h( \frac{tan \beta + tan \alpha }{tan \beta - tan \alpha } )$

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HOPE IT WILL HELP YOU ☺☺
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DEVIL_KING ▄︻̷̿┻̿═━一