a,b,c are sides of a triangle and are in GP. if log a - log 2b, log 2b -log 3c and log 3c - log a are in AP then

A)angle A must be obtuse      B) angle B must be obtuse 
C) C=90 degree                     D) none of these

pls give the full solution and not only the option

Dear student
Given:a,b,c are in GPb2=ac   ...1  If x,y,z are in GP y2=xzAlso  loga-log2b,log2b-log3c,log3c-loga are in A.P2log2b-log3c=  loga-log2b+log3c-loga  If x,y,z are in AP 2y=x+z2log2b-2log3c=log3c-log2b2log2b+log2b=log3c+2log3c3log2b=3log3clog2b=log3c2b=3cb=3c2 b2=9c24   and From 1, 9c24=acc=4a9Now putting the value of c=4a9 in  b2=9c24b2=94×16a281=4a29By cosine formula:cosA=b2+c2-a22bc=4a29+16a281-a22×2a3×4a9=36a2+16a2-81a28116a227=-2948i.e. cosA<0A>90°Hence obtuse

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