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
Answers
2hSeca/(Tanb - Tana) is the distance of the cloud from the point of observation
Step-by-step explanation:
Complete Question is :
the distance of the cloud from the point of observation is 2hSeca/(Tanb - Tana)
Let say Height of Cloud from Lake= C
Then Vertical height of Cloud from observation point = C - h
=> Vertical height of Cloud image in lake from observation point = C + h
Tana = (C - h)/( Horizontal Distance)
Tanb = (C + h)/( Horizontal Distance)
=> Tana/Tanb = (C - h)/(C + h)
=> CTana + hTana = CTanb - hTanb
=> C(Tanb - Tana) = h(Tana + Tanb)
=> C = h(Tana + Tanb)/(Tanb - Tana)
Sina = (C - h)/ the distance of the cloud from the point of observation
=> the distance of the cloud from the point of observation is = (C - h)/Sina
putting C = h(Tana + Tanb)/(Tanb - Tana)
= (h(Tana + Tanb)/(Tanb - Tana) - h)/Sina
= h(Tana + Tanb - Tanb + tana)/Sina(Tanb - Tana)
= 2hTana/Sina(Tanb - Tana)
= 2hSina/CosaSina(Tanb - Tana)
= 2h/Cosa(Tanb - Tana)
= 2hSeca/(Tanb - Tana)
QED
Proved
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