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Question
PQ is a post of given height a, and AB is a tower at some distance. If α and β are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.
Solution
Let AB be the tower of height H and PQ is a given post of height a, α and β are angles of elevation of the top of tower AB from P and Q. Let PA = x. PQ = a and BC = h.
The corresponding figure is as follows
In ΔQCB
`=> tan beta = h/x`
`=> x = h/(tan beta)
Again in ΔPAB
`=> tan alpha = (h + a)/x`
`=> tan alpha = ((h + a)tan beta)/h`
`=> h tan alpha = (h + a)tan beta`
`=> h(tan alpha - tan beta) = a tan beta`
`=> h =(a tan beta)/(tan alpha - tan beta)`
Now
`=> x = (a tan beta)/((tan alpha - tan beta) xx tan beta)`
`=> x = a/(tan alpha - tan beta)`
`=> H = (a tan alpha)/(tan a - tan beta)`
Hence required heigtht is `(a tan alpha)/(tan alpha - tan beta)` And distance is `a/(tan alpha - tan beta)`
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