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| 1. |
Two stations due south of a leaning tower |
| Answer» Let AB be the leaning tower and let C and D be two given stations at distances a and b respectively from the foot A of the tower.Let AE = x and BE = hIn\xa0{tex} \\triangle A E B,{/tex}\xa0we have{tex} \\tan \\theta = \\frac { B E } { A E }{/tex}{tex} \\Rightarrow \\quad \\tan \\theta = \\frac { h } { x }{/tex}{tex} \\Rightarrow \\quad x = h \\cot \\theta{/tex}\xa0.......(i)In {tex} \\Delta C E B,{/tex}\xa0we have{tex} \\tan \\alpha = \\frac { B E } { C E }{/tex}{tex} \\Rightarrow \\quad \\tan \\alpha = \\frac { h } { a + x }{/tex}{tex} \\Rightarrow \\quad a + x = h \\cot \\alpha{/tex}{tex} \\Rightarrow \\quad x = h \\cot \\alpha - a{/tex}.........(ii)In\xa0{tex} \\Delta D E B,{/tex}\xa0we have{tex} \\tan \\beta = \\frac { B E } { D E }{/tex}{tex} \\Rightarrow \\quad \\tan \\beta = \\frac { h } { b + x }{/tex}{tex} \\Rightarrow \\quad b + x = h \\cot \\beta{/tex}{tex} \\Rightarrow \\quad x = h \\cot \\beta - b{/tex}\xa0...........(iii)On equating the values of x obtained from equations (i) and (ii), we have{tex} h \\cot \\theta = h \\cot \\alpha - a{/tex}{tex} \\Rightarrow \\quad h ( \\cot \\alpha - \\cot \\theta ) = a{/tex}{tex} \\Rightarrow \\quad h = \\frac { a } { \\cot \\alpha - \\cot \\theta }{/tex} .........(iv)On equating the values of x obtained from equations (i) and (iii), we get{tex} h \\cot \\theta = h \\cot \\beta - b{/tex}{tex} \\Rightarrow \\quad h ( \\cot \\beta - \\cot \\theta ) = b{/tex}{tex} \\Rightarrow \\quad h = \\frac { b } { \\cot \\beta - \\cot \\theta }{/tex} .........(v)Equating the values of h from equations (iv) and (v), we get{tex} \\frac { a } { \\cot \\alpha - \\cot \\theta } = \\frac { b } { \\cot \\beta - \\cot \\theta }{/tex}{tex} \\Rightarrow \\quad a ( \\cot \\beta - a \\cot \\theta ) = b ( \\cot \\alpha - \\cot \\theta ){/tex}{tex} \\Rightarrow \\quad ( b - a ) \\cot \\theta = b \\cot \\alpha - a \\cot \\beta{/tex}{tex} \\Rightarrow \\quad \\cot \\theta = \\frac { b \\cot \\alpha - a \\cot \\beta } { b - a }{/tex} | |