Let h1,h2,h3 are lengths of altitudes from the vertices A,B,C of triangle ABC respectively. Let M be a point inside the triangle ABC,Whose distances from BC,CA,AB are respectively equal to u,v,w then the least
Value of h1/u+h2/v+h3/w=………
1)3.     2)6.      3)9.    4)12
 

Asked by Sunil Soni | 7th Dec, 2015, 11:34: PM

Expert Answer:

G i v e n space t h a t space h subscript 1 comma h subscript 2 comma h subscript 3 space a r e space t h e space a l t i t u d e s space f r o m space t h e space v e r t i c e s space o f space t h e space t r i a n g l e space triangle A B C. L e t space M space b e space a n space i n t e r i o r space p o i n t space a n d space w h o s e space d i s tan c e s space f r o m space B C comma space C A space a n d space A B space a r e space r e s p e c t i v e l y space e q u a l space t o space u comma space v space a n d space w. N o w space c o n s i d e r space t h e space n u m b e r s comma space u over h subscript 1 comma space v over h subscript 2 space a n d space w over h subscript 3 T h e space a r i t h m e t i c space m e a n space o f space t h e space a b o v e space t h r e e space n u m b e r s equals fraction numerator u over h subscript 1 plus v over h subscript 2 plus w over h subscript 3 over denominator 3 end fraction T h e space h a r m o n i c space m e a n space o f space t h e space a b o v e space t h r e e space n u m b e r s equals fraction numerator 3 over denominator begin display style h subscript 1 over u end style plus begin display style h subscript 2 over v end style plus begin display style h subscript 3 over w end style end fraction 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 o f space a n y space t h r e e space n u m b e r s space i s space a l w a y s space g r e a t e r space t h a n space h a r m o n i c space m e a n space o f space t h e space s a m e space n u m b e r s. T h u s comma space fraction numerator u over h subscript 1 plus v over h subscript 2 plus w over h subscript 3 over denominator 3 end fraction greater or equal than fraction numerator 3 over denominator begin display style h subscript 1 over u plus plus h subscript 2 over v h subscript 3 over w end style end fraction... left parenthesis 1 right parenthesis A l s o space sin c e space M space i s space t h e space i n t e r i o r space p o i n t comma space w e space k n o w space t h a t space u over h subscript 1 plus v over h subscript 2 plus w over h subscript 3 equals 1 S u b s t i t u t i n g space t h e space v a l u e space u over h subscript 1 plus v over h subscript 2 plus w over h subscript 3 equals 1 space i n space t h e space i n e q u a l i t y space left parenthesis 1 right parenthesis comma space w e space h a v e comma 1 third greater or equal than fraction numerator 3 over denominator begin display style h subscript 1 over u plus h subscript 2 over v plus h subscript 3 over w end style end fraction rightwards double arrow h subscript 1 over u plus h subscript 2 over v plus h subscript 3 over w greater or equal than 3 cross times 3 rightwards double arrow h subscript 1 over u plus h subscript 2 over v plus h subscript 3 over w greater or equal than 9 T h u s space t h e space l e a s t space v a l u e space o f space h subscript 1 over u plus h subscript 2 over v plus h subscript 3 over w space i s space 9 space a n d space h e n c e space left parenthesis 3 right parenthesis space i s space t h e space c o r r e c t space o p t i o n.

Answered by Vimala Ramamurthy | 9th Dec, 2015, 11:41: AM