Question

If the sides of a triangle are in GP and the largest angle is twice the smallest angle, then the common ratio, which is greater than 1, lies in the interval

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

$a,b=ar,c=a{r}^2$, where $r>1$ form the question c=2A
So, $B=\pi -A-c=\pi -3A$
$\therefore \frac{a}{sinA}=\frac{b}{sin3A}=\frac{c}{sin2A}\Rightarrow \frac{1}{sinA}=\frac{r}{sin3A}=\frac{r^2}{sin2A}$
$\therefore r^2=2cosAandr=\frac{sin3A}{sinA}=3-4si{n}^{2}A=4co{s}^{2}A-1$
$\Rightarrow 0<r<\sqrt{2}$

$\Rightarrow$ option B is correct