MATHS - CLASS X
[tex]The \: angle \: of \: elevation \: of \: a \: cloud \\ from \: a \: point \: h \: metres \: above \: the \\ surface \: of \: a \: lake \: is \: \theta \: and \: the \: angle \\ of \: depression \: of \: its \: reflection \\ in \: the \: lake \: is \: \alpha . \: Prove \: that \: the \\ height \: of \: the \: cloud \: above \: the \: lake \\ is \: h( \frac{ \tan\alpha + \tan \theta }{ \tan\alpha - \tan \theta } ).[/tex]
Answers & Comments
Let C be the Position of the Cloud
A-B Line Represents the Line which is 'h' metres above the Lake.
A is the Point from which the Cloud is Observed and θ is the Angle of Elevation of the Cloud viewed from Point A
C'' is the Reflection of the Cloud seen from Point A
α is the Angle of Depression of the Cloud reflection viewed from Point A
LK is the Surface of the Lake
z is the Height of the Cloud from the Point where it is Viewed (i.e.) AB Line.
From the Figure :
⇒![AB = \frac{z}{Tan(theta)} AB = \frac{z}{Tan(theta)}](https://tex.z-dn.net/?f=AB%20%3D%20%5Cfrac%7Bz%7D%7BTan%28theta%29%7D)
In the Similar way :
⇒![AB = \frac{2h + z}{Tan\alpha} AB = \frac{2h + z}{Tan\alpha}](https://tex.z-dn.net/?f=AB%20%3D%20%5Cfrac%7B2h%20%2B%20z%7D%7BTan%5Calpha%7D)
Equating Both 'AB's
⇒![\frac{z}{Tan(theta)} = \frac{2h + z}{Tan\alpha} \frac{z}{Tan(theta)} = \frac{2h + z}{Tan\alpha}](https://tex.z-dn.net/?f=%5Cfrac%7Bz%7D%7BTan%28theta%29%7D%20%3D%20%5Cfrac%7B2h%20%2B%20z%7D%7BTan%5Calpha%7D)
⇒ z × Tanα = 2h × Tanθ + z × Tanθ
⇒ z(Tanα - Tanθ) = 2hTanθ
⇒![z = \frac{2hTan(theta)}{Tan\alpha - Tan(theta)} z = \frac{2hTan(theta)}{Tan\alpha - Tan(theta)}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B2hTan%28theta%29%7D%7BTan%5Calpha%20-%20Tan%28theta%29%7D)
But the Height of the Cloud above the Lake is z + h
⇒![z + h = \frac{2hTan(theta)}{Tan\alpha - Tan(theta)} + h z + h = \frac{2hTan(theta)}{Tan\alpha - Tan(theta)} + h](https://tex.z-dn.net/?f=z%20%2B%20h%20%3D%20%5Cfrac%7B2hTan%28theta%29%7D%7BTan%5Calpha%20-%20Tan%28theta%29%7D%20%2B%20h)
⇒![z + h = \frac{2hTan(theta) - hTan(theta) + hTan\alpha}{Tan\alpha - Tan(theta)} z + h = \frac{2hTan(theta) - hTan(theta) + hTan\alpha}{Tan\alpha - Tan(theta)}](https://tex.z-dn.net/?f=z%20%2B%20h%20%3D%20%5Cfrac%7B2hTan%28theta%29%20-%20hTan%28theta%29%20%2B%20hTan%5Calpha%7D%7BTan%5Calpha%20-%20Tan%28theta%29%7D)
⇒![z + h = \frac{h(Tan\alpha + Tan(theta))}{Tan\alpha - Tan(theta)} z + h = \frac{h(Tan\alpha + Tan(theta))}{Tan\alpha - Tan(theta)}](https://tex.z-dn.net/?f=z%20%2B%20h%20%3D%20%5Cfrac%7Bh%28Tan%5Calpha%20%2B%20Tan%28theta%29%29%7D%7BTan%5Calpha%20-%20Tan%28theta%29%7D)