If a, b,c,d are in G.P.  then the value of  (ab+bc+cd)2 - (a2+b2+c2)(b2+c2+d2) is
 
a) 0     b)  1   c) -1    d) None
 

Asked by ahuja8087 | 30th Oct, 2015, 11:55: PM

Expert Answer:

G i v e n space t h a t space a comma b comma c comma space a n d space d space a r e space i n space G P. space S i n c e space a comma b space a n d space c space a r e space i n space G P comma space w e space h a v e comma space a cross times c equals b squared... left parenthesis 1 right parenthesis S i n c e space b comma c space a n d space d space a r e space i n space G P comma space w e space h a v e comma space b cross times d equals c squared... left parenthesis 2 right parenthesis D i v i d i n g space t h e space e q u a t i o n space left parenthesis 1 right parenthesis space b y space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma fraction numerator a cross times c over denominator c squared end fraction equals fraction numerator b squared over denominator b cross times d end fraction rightwards double arrow a over c equals b over d rightwards double arrow a d equals b c... left parenthesis 3 right parenthesis 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 a d close parentheses squared equals open parentheses b c close parentheses squared rightwards double arrow a squared d squared equals b squared c squared... left parenthesis 4 right parenthesis rightwards double arrow a cross times d cross times a cross times d equals b squared c squared rightwards double arrow a cross times b cross times c cross times d equals b squared c squared space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 3 right parenthesis close square brackets... left parenthesis 5 right parenthesis R e w r i t i n g space e q u a t i o n space left parenthesis 4 right parenthesis comma space w e space h a v e comma rightwards double arrow a squared d squared equals b cross times c cross times b cross times c rightwards double arrow a squared d squared equals a cross times b cross times c cross times d space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 3 right parenthesis close square brackets... left parenthesis 6 right parenthesis space  N o w space c o n s i d e r space t h e space e x p r e s s i o n a squared cross times c squared equals a cross times c cross times a cross times c space space space space space space space space space equals a cross times c cross times b squared space space space space space space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets therefore a squared cross times c squared equals a cross times c cross times b squared space space space space space space space space space space space... left parenthesis 7 right parenthesis  A l s o comma space w e space h a v e comma b squared cross times d squared equals b cross times d cross times b cross times d space space space space space space space space space equals b cross times d cross times c squared space space space space space space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 2 right parenthesis close square brackets therefore b squared cross times d squared equals b cross times d cross times c squared space space space space space space space space space space space... left parenthesis 8 right parenthesis  N o w space c o n s i d e r space t h e space e x p r e s s i o n b to the power of 4 equals b squared cross times b squared space space space equals b squared cross times a cross times c space space space space space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets therefore b to the power of 4 equals b squared cross times a cross times c space space space space space space space space space space space... left parenthesis 9 right parenthesis  A l s o comma space w e space h a v e comma c to the power of 4 equals c squared cross times c squared space space space equals c squared cross times b cross times d space space space space space space space space space space open square brackets because f r o m space e q u a t i o n space left parenthesis 2 right parenthesis close square brackets therefore c to the power of 4 equals c squared cross times b cross times d space space space space space space space space space space... left parenthesis 10 right parenthesis N o w space c o n s i d e r space t h e space e x p r e s s i o n open parentheses a b plus b c plus c d close parentheses squared minus open parentheses a squared plus b squared plus c squared close parentheses open parentheses b squared plus c squared plus d squared close parentheses equals a squared b squared plus b squared c squared plus c squared d squared plus 2 a b squared c plus 2 b c squared d plus 2 a b c d minus a squared b squared space space space space space minus a squared c squared minus a squared d squared minus b to the power of 4 minus b squared c squared minus b squared d squared minus b squared c squared minus c to the power of 4 minus c squared d squared space space space space space space equals 2 a b squared c plus 2 b c squared d plus 2 a b c d minus a squared c squared minus a squared d squared minus b to the power of 4 minus b squared d squared minus b squared c squared minus c to the power of 4 space space space space space open square brackets because f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis close square brackets equals 2 b to the power of 4 plus 2 c to the power of 4 plus b squared c squared plus a squared d squared minus a squared c squared minus a squared d squared minus b to the power of 4 minus b squared d squared minus b squared c squared minus c to the power of 4 space space space space space space open square brackets because f r o m space e q u a t i o n s space left parenthesis 3 right parenthesis comma space left parenthesis 4 right parenthesis comma space left parenthesis 5 right parenthesis comma space left parenthesis 6 right parenthesis comma space left parenthesis 7 right parenthesis space a n d space left parenthesis 8 right parenthesis close square brackets equals b to the power of 4 plus c to the power of 4 minus a squared c squared minus b squared d squared equals b to the power of 4 plus c to the power of 4 minus b to the power of 4 minus c to the power of 4 space space space space left square bracket because f r o m space e q u a t i o n s space left parenthesis 9 right parenthesis space a n d space left parenthesis 10 right parenthesis right square bracket equals 0 H e n c e space t h e space c o r r e c t space o p t i o n space i s space left parenthesis a right parenthesis.

Answered by Vimala Ramamurthy | 31st Oct, 2015, 05:12: PM