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Question

# If a,b,c are in G.P. and logaālog2b,log2bā log3c and log3cāloga are in A.P., then a,b,c are the length of the sides of a triangle which is

A
Acute angled
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B
Obtuse angled
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C
Right angled
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D
Equilateral
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Solution

## The correct option is B Obtuse angledAs given b2=ac and 2(log2b−log3c)=loga−log2b+log3c−loga ⇒b2=acand2b=3c⇒b=2a3 and 4a9 Since a+b=5a3>c,b+c=10a9,>a,c+a=13a9>b It implies that a,b,c from a triangle with a as the greatest side. Now, let us find the greatest angle A of △ABC by using the cosine formula. cosA=b2+c2−a22bc=−2948<0 ∴ The angle A is obtuse.

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