A tree standing on horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, angles of elevation of the top are respectively `alpha` and `beta`. Prove that height of the top from the ground is `((b-a).tanalpha.tanbeta)/(tanalpha-tanbeta)`

A tree standing on horizontal plane is leaning towards east. At two po

1.	A tree standing on horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, angles of elevation of the top are respectively `alpha` and `beta`. Prove that height of the top from the ground is `((b-a).tanalpha.tanbeta)/(tanalpha-tanbeta)`
Answer» We can draw a diagram with the given details. Please refer to video for the diagram. Here, height of top from the ground ` = OB = h` `AC = a, AD = b, OB =x` `/_OCB = alpha, /_ODB = beta` `:.tan alpha = (OB)/(OC) ` `=> tan alpha = h/(x+a)` `h = tan alpha(x+a)->(1)` `tan beta = (OB)/(OD) ` `=> tan beta = h/(x+b)` `=> tan beta(x+b) = h` Putting value of `h` from (1), `=>tan alpha (x+a) = tan beta(x+b)` `=>x(tan alpha- tan beta) = btanbeta - atanalpha` `=>x = ( btanbeta - atanalpha)/(tan alpha- tan beta)` `:. h = tan alpha(( btanbeta - atanalpha)/(tan alpha- tan beta) +a)` `=> h = tan alpha(( btanbeta - atanalpha+atanalpha-atanbeta)/(tan alpha- tan beta))` `=> h =( (b-a)tan beta tanalpha)/(tan alpha- tan beta)`

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A tree standing on horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it, angles of elevation of the top are respectively `alpha` and `beta`. Prove that height of the top from the ground is `((b-a).tanalpha.tanbeta)/(tanalpha-tanbeta)`

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