If `alpha, beta and gamma` are three consecutive terms of a non-constant G.P. such that the equations `ax^(2) + 2beta x + gamma = 0 and x^(2) + x - 1= 0` have a common root, then `alpha (beta + gamma)` is equal to

Options

A. `beta gamma`
B. 0
C. `alpha beta`
D. `alpha gamma`

Answer

A

Explanation

`beta^(2)=alpha gamma , beta =sqrt(alphagamma)`
`(sqrt(alpha x) + sqrt(gamma))^(2)=0impliesx=-(sqrt(gamma))/(sqrt(alpha)0` is a root of `x^(2)+x-1=0implies(gamma)/(alpha)-(sqrt(gamma))/(sqrt(alpha))-1=0`
`implies (gamma-sqrt(alpha gamma))/(alpha)=1implies((gamma-beta))/(alpha)=1impliesgamma=(beta+alpha)`
`alpha beta+alpha gamma= alpha beta+beta^(2)=beta(alpha+beta)=beta gamma`