If a,b,c are 3 distinct terms in G.P. and are also the lengths of sides of a triangle, then number of integral values of the common ratio of that G.P. is:

Options

A. 0
B. 1
C. 2
D. Infinitely many

Answer

A

Explanation

Let this sides by ` a, ar, ar^2`, we must have `a > 0, r > 0 and r != 0` and
`a + ar > ar^2 implies r^2 - r - 1 < 0 implies r in ((1 - sqrt(5))/2, (1 + sqrt(5))/(2))`
`a + ar^2 > ar implies r^2 - r + 1 > 0 implies r in R`
`ar + ar^2 > a implies r^2 + r - 1 > 0 implies r in (-oo, (-1,-sqrt(5))/(2)) uu ((sqrt(5)-1)/(2), oo)`
Hence `r in ((sqrt(5)-1)/2, (sqrt(5)+1)/(2))-{1}`.