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Question

The sides a,b,c, of ΔABC are in G.P, and loga - log2b, log2b - log3c, log3c - loga are in A.P., then ΔABC

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

The correct option is C Acute angled

Given that, In ΔABC

Sides a,b,c in G.P.

Then,

ba=cb …….. (1)

b2=ac …… (2)

Now given that,

logalog2b,log2blog3c,log3cloga in A.P.

Then, we know that In an A.P.

(log2blog3c)(logalog2b)=(log3cloga)(log2blog3c)

log2b+log2blog3cloga=log3c+log3clogalog2b

2log2blog3cloga=2log3clogalog2b

2log2b+log2b=log3c+2log3c

3log2b=3log3c

log2b=log3c

Eliminating log both side and we get,

2b=3c

cb=23

By equation (1)

ba=cb=23

Then,

b=23a and c=23b

By equation (2),

b2=ac put b=23a and we get

(2a3)2=ac

c=4a9

Now, a,b=2a3,c=4a9

Using cos rule and we gat

cosA=b2+c2a22bc

cosA=(2a3)2+(4a9)2a22×2a3×4a9

cosA=4a92+16a812a22×2a3×4a9

cosA=2948=12

cosA=cos(450)

cosA=cos1350

And A=1350


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