Question

If a,b,c are in G.P. and $\log a-\log 2b,\log 2b-log3c$ and $\log 3c-\log a$ are in A.P., then a,b,c are the length of the sides of a triangle which is

• A. Acute angled
• B. Obtuse angled
• C. Right angled
• D. Equilateral

Answer 1

The correct option is B Obtuse angled

As given $b^2=ac$ and
$2(\log 2b-\log 3c)=\log a-\log 2b+\log 3c-\log a$
$\Rightarrow b^2=acand2b=3c\Rightarrow b=\frac{2a}{3}$ and $\frac{4a}{9}$
Since $a+b=\frac{5a}{3}>c,b+c=\frac{10a}{9},>a,c+a=\frac{13a}{9}>b$
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.
$\cos A=\frac{b^2+c^2-a^2}{2bc}=-\frac{29}{48}<0$
$\therefore$ The angle A is obtuse.