If the sides of a triangle are in G.P., and its largest angle is twice the smallest, then the common ratio `r` satisfies the inequality `0

Options

A. `0 lt r lt sqrt(2)`
B. `1 lt r lt sqrt(2)`
C. `1 lt r lt 2`
D. none of these

Answer

B

Explanation

Let the sides of the triangle be a/r,a and ar, with `agt0` and `rgt1`.
Let `alpha` be the smallest angle, so that the largest angleis `2alpha`. Then `alpha` is opposite to the side a/r, and `2alpha` is positive to the side ar. Applying sine rule, we get
`(a//r)/(sin alpha)=(ar)/(ain2alpha)`
or `(sin2alpha)/(sinalpha)=r^(2)`
or `2cosalpha=r^(2)`
`rArrr^(2)lt2`
or `rltsqrt2`
Hence, `1ltrltsqrt2`