A round ball of radius r subtends an angle alpha at  the eye  of the observer while the angle of elevation of its centre is beta.Prove that the height of the centre of the balloon is r sinbold italic beta bold space bold italic c bold italic o bold italic s bold italic e bold italic c bold space bold italic alpha bold divided by bold 2

Asked by afreensyed13 | 14th Feb, 2017, 07:57: AM

Expert Answer:

begin mathsize 16px style To space prove colon space height space equals DB equals straight r space sinβ space cosec straight alpha over 2
In space increment ACD space and space AED comma
AC equals AE space... left parenthesis tangents space drawn space from space an space external space point space to space the space circle space are space equal right parenthesis
AD equals AD space... left parenthesis common space side right parenthesis
CD equals DE equals straight r space... left parenthesis radii space of space the space same space circle right parenthesis
rightwards double arrow increment ACD space approximately equal to space AED space... left parenthesis SSS space congruence space criterion right parenthesis
So comma space angle CAD equals angle DAE equals straight alpha over 2 space... left parenthesis cpct right parenthesis
In space increment ACD comma
sin straight alpha over 2 equals space CD over AD rightwards double arrow sin straight alpha over 2 equals space straight r over AD rightwards double arrow AD equals fraction numerator straight r space over denominator sin straight alpha over 2 end fraction rightwards double arrow AD equals straight r space cosec space straight alpha over 2.... left parenthesis straight i right parenthesis
In space increment DAB comma
sin space straight beta equals DB over AD rightwards double arrow AD equals fraction numerator DB over denominator sin space straight beta end fraction rightwards double arrow AD equals fraction numerator DB over denominator sin space straight beta end fraction.... left parenthesis ii right parenthesis
From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
straight r space cosec space straight alpha over 2 equals fraction numerator DB over denominator sin space straight beta end fraction rightwards double arrow DB equals straight r space cosec space straight alpha over 2 space sin space straight beta rightwards double arrow DB equals straight r space sin space straight beta space cosec space straight alpha over 2
So comma space the space height equals straight r space sin space straight beta space cosec space straight alpha over 2
Hence space proved. end style

Answered by Rebecca Fernandes | 14th Feb, 2017, 10:32: AM

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