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CBSE Class 12-science Answered

If a2 , b2 , c2 are in ap prove that a/b+c , b/c+a , c/a+b are also in ap and 1/b+c , 1/c+a, 1/a+b are also in ap
Asked by rushabhjain.avv | 19 Nov, 2018, 08:44: PM
answered-by-expert Expert Answer
Let us consider the three terms :  begin mathsize 12px style fraction numerator a over denominator b plus c end fraction space comma space fraction numerator b over denominator c plus a end fraction space comma space fraction numerator c over denominator a plus b end fraction end style
Difference between 2nd and first term is worked out as follows
begin mathsize 12px style fraction numerator b over denominator c plus a end fraction space minus space fraction numerator a over denominator b plus c end fraction space equals space fraction numerator open parentheses b minus a close parentheses begin display style open parentheses a plus b plus c close parentheses end style over denominator open parentheses b plus c close parentheses begin display style open parentheses c plus a close parentheses end style end fraction space............. left parenthesis 1 right parenthesis sin c e space a squared space comma space b squared space a n d space c squared space a r e space i n space A. P comma space b squared space minus space a squared space equals space d space space o r space left parenthesis b minus a right parenthesis space equals space fraction numerator d over denominator left parenthesis a plus b right parenthesis end fraction space space........... left parenthesis 2 right parenthesis b y space s u b s t i t u t i n g space f o r space left parenthesis b minus a right parenthesis space u sin g space left parenthesis 2 right parenthesis space i n space e q n. left parenthesis 1 right parenthesis comma space w e space g e t space fraction numerator begin display style b end style over denominator begin display style c plus a end style end fraction space minus space fraction numerator begin display style a end style over denominator begin display style b plus c end style end fraction space equals fraction numerator begin display style d open parentheses a plus b plus c close parentheses end style over denominator begin display style left parenthesis end style begin display style a end style begin display style plus end style begin display style b end style begin display style right parenthesis end style open parentheses b plus c close parentheses begin display style open parentheses c plus a close parentheses end style end fraction space.............. left parenthesis 3 right parenthesis S i m i l a r l y space w e space c a n space p r o v e space t h e space d i f f e r e n c e space b e t w e e n space t h i r d space a n d space s e c o n space t e r m space a s comma space space fraction numerator begin display style c end style over denominator begin display style a plus b end style end fraction space minus space fraction numerator begin display style b end style over denominator begin display style c plus a end style end fraction space equals fraction numerator begin display style d open parentheses a plus b plus c close parentheses end style over denominator begin display style left parenthesis a plus b right parenthesis open parentheses b plus c close parentheses open parentheses c plus a close parentheses end style end fraction space.......... left parenthesis 4 right parenthesis F r o m space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space i t space w i l l space b e space k n o w n space t h a t space fraction numerator a over denominator b plus c end fraction space comma space fraction numerator b over denominator c plus a end fraction space a n d space fraction numerator c over denominator a plus b end fraction space a r e space i n space A. P end style
 
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Now we see the terms,  begin mathsize 12px style fraction numerator 1 over denominator b plus c end fraction space comma space fraction numerator 1 over denominator c plus a end fraction space space a n d space fraction numerator 1 over denominator a plus b end fraction end style
difference between second and first term is worked out as follows
begin mathsize 12px style fraction numerator 1 over denominator c plus a end fraction space minus space fraction numerator 1 over denominator b plus c end fraction space equals space fraction numerator open parentheses b minus a close parentheses over denominator open parentheses c plus a close parentheses begin display style open parentheses b plus c close parentheses end style end fraction equals fraction numerator d over denominator open parentheses a plus b close parentheses open parentheses b plus c close parentheses begin display style open parentheses c plus a close parentheses end style end fraction space............... left parenthesis 5 right parenthesis a s space b e f o r e space w e space u s e d space d space equals space left parenthesis b squared minus a squared right parenthesis space i n space a b o v e space e q n. S i m i l a r l y space i t space c a n space b e space p r o v e d space f o r space t h e space d i f f e r e n c e space o f space t h i r d space a n d space s e c o n d space t e r m space fraction numerator begin display style 1 end style over denominator begin display style a plus b end style end fraction space minus space fraction numerator begin display style 1 end style over denominator begin display style c plus a end style end fraction space equals space fraction numerator d over denominator open parentheses a plus b close parentheses open parentheses b plus c close parentheses begin display style open parentheses c plus a close parentheses end style end fraction space space............... left parenthesis 6 right parenthesis end style
from (5) and (6), we say that begin mathsize 12px style fraction numerator 1 over denominator b plus c end fraction space comma space fraction numerator 1 over denominator c plus a end fraction space space a n d space fraction numerator 1 over denominator a plus b end fraction end style are in A.P
Answered by Thiyagarajan K | 20 Nov, 2018, 05:42: AM

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