In a triangleABC, bisector of angle C meets the side AB at D
circumcircle at E.The maximum value of CD.DE is equal to
 
1) a^2/4    2) b^2/4.   3) c^2/4.    4)(a+by)^2/4

Asked by Sunil Soni | 7th Dec, 2015, 10:43: PM

Expert Answer:

L e t space A B C space b e space a space t r i a n g l e. space G i v e n space t h a t space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space a n g l e space C space m e e t s space t h e space s i d e space A B space a t space D space a n d space t h e space c i r c u m c i r c l e space a t space E. O b s e r v e space t h e space f o l l o w i n g space f i g u r e.
I f space t w o space c h o r d s space A B space a n d space C D space i n t e r s e c t space a t space P space i n s i d e space t h e space c i r c l e space w i t h space c e n t r e space a t space O comma space t h e n space P A cross times P B space equals space P C cross times P D. space H e r e comma space i n space t h e space a b o v e space d i a g r a m comma space A B space a n d space C E space i n t e r s e c t space a t space D. T h e r e f o r e comma space A D cross times D B equals C D cross times D E. comma.. left parenthesis 1 right parenthesis S i n c e space A B equals c space i s space t h e space s i d e space o f space t h e space t r i a n g l e comma space w e space h a v e comma space A D plus D B equals A B equals c... left parenthesis 2 right parenthesis W e space k n o w space t h a t space a r i t h m e t i c space m e a n space i s space a l w a y s space g r e a t e r space t h a n space t h e space g e o m e t r i c space m e a n. T h u s comma fraction numerator A D plus D B over denominator 2 end fraction greater or equal than square root of A D cross times D B end root S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma open parentheses fraction numerator A D plus D B over denominator 2 end fraction close parentheses squared greater or equal than open parentheses square root of A D cross times D B end root close parentheses squared rightwards double arrow open parentheses A D plus D B close parentheses squared over 4 greater or equal than A D cross times D B rightwards double arrow c squared over 4 greater or equal than A D cross times D B... open square brackets f r o m space e q u a t i o n space left parenthesis 2 right parenthesis close square brackets rightwards double arrow c squared over 4 greater or equal than C D cross times D E.... 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

Answered by Vimala Ramamurthy | 8th Dec, 2015, 11:50: AM