Two points at a distance "a" are on the east and west of a tower.The angles of elevation of the top of the tower from these points are alpha space a n d space beta respectively.Prove that the height of the tower is  fraction numerator a over denominator c o t alpha plus c o t beta end fraction

Asked by afreensyed13 | 13th Feb, 2017, 11:15: AM

Expert Answer:

 
 
 
begin mathsize 16px style CD space is space the space tower. space The space two space points space on space the space east space and space west space of space the space tower space are space straight A space and space straight B space respectively.
Given space that space the space points space are space at space straight a space distance space straight a space from space eachother space and space are space on space either space sides space left parenthesis that space is space east space and space west right parenthesis space of space the space tower.
Let space AD equals straight x space and space hence space DB equals AB minus AD equals straight a minus straight x

In space increment ADC comma space
cot space straight alpha equals straight x over straight h rightwards double arrow straight x equals straight h space cot space straight alpha space.... left parenthesis straight i right parenthesis
In space increment BDC comma
cot space straight beta equals fraction numerator straight a minus straight x over denominator straight h end fraction rightwards double arrow straight a minus straight x equals straight h space cot space straight beta space...... left parenthesis ii right parenthesis
Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
straight x plus straight a minus straight x equals straight h space cot space straight alpha plus straight h space cot space straight beta
rightwards double arrow straight a equals straight h space left parenthesis cot space straight alpha space plus space cot space straight beta right parenthesis
rightwards double arrow straight h equals fraction numerator straight a over denominator cot space straight alpha space plus space cot end fraction
Hence comma space the space height space of space the space tower space is space fraction numerator straight a over denominator cot space straight alpha space plus space cot end fraction. end style
 
Hi, kindly note that we are not given that the points A and B are not given to be at an equal distance from the tower. Hence I have taken AD = x and DB = a - x.
Let us know if you have any doubt regarding this. Thank you,

Answered by Rebecca Fernandes | 12th Mar, 2017, 03:36: PM

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