A round balloon of radius 'a' subtends an angle theta at the eye of the observer while the angle of elevation of its center is ϕ. Prove that the height of the center of the balloon is
 asin ϕcosec theta/2. please send the solution with the figure.
 
 
 
 
 
 
 
 
 
 

Asked by joshiraghavendra53 | 18th Nov, 2014, 08:53: 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 | 19th Nov, 2014, 09:56: AM