A ROUND BALOON 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 BALOON IS rsinbetacosecalpha/2

Asked by apporv1999 | 30th Oct, 2014, 07:16: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space angle A P B equals alpha T h e space l e n g t h s space o f space t h e space tan g e n t s space f r o m space a n space e x t e r n a l space p o i n t space a r e space e q u a l. triangle A P O space a n d space triangle B P O space a r e space c o n g r u e n t. H e n c e comma space angle A P O equals angle B P O equals alpha over 2 C o n s i d e r space t h e space t r i a n g l e space triangle A P O : sin alpha over 2 equals fraction numerator O A over denominator O P end fraction rightwards double arrow O P equals fraction numerator O A over denominator sin alpha over 2 end fraction rightwards double arrow O P equals r cos e c alpha over 2.... left parenthesis 1 right parenthesis N o w space c o n s i d e r space t h e space t r i a n g l e comma space triangle O P G : sin beta equals fraction numerator O G over denominator O P end fraction rightwards double arrow O G equals O P sin beta rightwards double arrow O G equals r cos e c alpha over 2 sin beta space space space space space space space open square brackets f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets T h u s comma space h e i g h t space o f space t h e space c e n t r e space o f space t h e space b a l l o o n space i s space r sin beta cos e c alpha over 2
 

Answered by Vimala Ramamurthy | 30th Oct, 2014, 11:11: PM