If the equation `z^4+a_1z^3+a_2z^2+a_3z+a_4=0` where `a_1,a_2,a_3,a_4` are real coefficients different from zero has a pure imaginary root then the expression `(a_1)/(a_1a_2)+(a_1a_4)/(a_2a_3)` has the value equal to

Options

A. 0
B. 1
C. -2
D. 2

Answer

B

Explanation

Let xi be the root where `x ne 0` and `x in R`
`x^(4) - a_(1)x^(3)I - a_(3)x^(2) + a_(3)xi + a_(4) = 0" "(1)`
`rArr x^(4) - a_(2)x^(2) + a_(4) = " "(2)`
and `a_(1)x^(3) - a_(3)x =0`
Form Eq. (2),
`a_(1)x^(3) - a_(3) = 0`
`rArr x^(2) = a_(3)//a_(1)` (as `x ne 0`)
Putting the value of `x^(2)` in Eq.(1), we get
`(a_(3)^(2))/(a_(1)^(2))- (a_(2)a_(3))/(a_(1))+a_(4) = 0`
`rArr a_(3)^(2) + a_(4) a_(1)a_(3)`
`rArr (a_(3))/(a_(1)a_(2))+(a_(1)a_(4))/(a_(2)a_(3))= 1` (dividing by `a_(1)a_(2)a_(3))`