Suppose the sides of a triangle form a geometric progression with common ratio r. Then r lies in the interval-

Options

A. `(0,(-1+sqrt(5))/(2))`
B. `((1+sqrt(5))/(2),(2+sqrt(5))/(2))`
C. `((-1+sqrt(5))/(2),(1+sqrt(5))/(2))`
D. `((2+sqrt(5))/(2),oo)`

Answer

C

Explanation

`a + ar gt ar^(2)`
` r^(2) - r - 1 lt 0`
`r in ((1-sqrt(5))/(2), (1+sqrt(5))/(2))"…….."(1)`
`ar^(2) + ar gt 0`
`r gt (-1 + sqrt(5))/(2) , r lt (-1-sqrt(5))/(2)"….."(2)`
`ar^(2) + a gt ar , r^(2) - 1 + 1 gt 0` always true
Solving (1) & (2)
`r in ((sqrt(5) - 1)/(2), (sqrt(5) + 1)/(2))`