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Question

If a,b,c are in G.P. and logalog2b,log2b log3c and log3cloga 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 angled
As given b2=ac and
2(log2blog3c)=logalog2b+log3cloga
b2=acand2b=3cb=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+c2a22bc=2948<0
The angle A is obtuse.

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