Question

If the equation $z^4+a_1{z}^3+a_2{z}^2+a_{3}z+a_4=0$, where $a_1,a_2,a_3,a_4$ are coefficients different from zero, has a purely imaginary roots, then the value the expression $\frac{a_3}{\left(a_1{a}_2\right)}+\frac{\left(a_1{a}_4\right)}{\left(a_2{a}_3\right)}$ has the value equal to

• A. 0
• B. 1
• C. -2
• D. 2

Answer 1

Answer: (B)

1

$z^4+a_1{z}^3+a_2{z}^2+a_{3}z+a_4=0$
Let $w$ be a purely imaginary root.
$⟹w=-\overline{w}$
$⟹w^4+a_1{w}^3+a_2{w}^2+a_{3}w+a_4=0$ ...(1)
Conjugating equation (1)
${\overline{w}}^4+a_1{\overline{w}}^3+a_2{\overline{w}}^2+a_3\overline{w}+a_4=0$
$⟹w^4-a_1{w}^3+a_2{w}^2-a_{3}w+a_4=0$ ...(2)
Adding eq. (1) & eq. (2)
$⟹2w^4+2a_2{w}^2+2a_4=0$
$⟹w^4+a_2{w}^2+a_4=0$ ...(3)
Subtracting eq. (1) & eq. (2)
$⟹2a_1{w}^3+2a_{3}w=0$
$⟹w^2=\frac{-a_3}{a_1}$ ...(4)

Combining eq. (3) & eq. (4) we get,
${\left(\frac{a_3}{a_1}\right)}^2-\frac{a_2{a}_3}{a_1}+a_4=0$
$⟹\frac{a_3}{a_1{a}_2}+\frac{a_4{a}_1}{a_2{a}_3}=1$
Ans: B