If z is a complex number satisfying z2+(3/z2)=-4 , then the sum of the absolute values of the imaginary parts of the roots of the equation is

Question

If z is a complex number satisfying z2+(3/z2)=-4 , then the sum of the absolute values of the imaginary parts of the roots of the equation is (A) 0 (B) 2√3 (C) 2 (D) 2√3+2

Tardigrade Admin · Accepted Answer

z4+4z2+3=0 (z2 + 1)(z2 + 3)=0 ⇒ z2=-1 or -3 ⇒ z=± i,±√3i Hence, the required sum is |1|+|- 1|+|√3|+|- √3|

Q. If $z$ is a complex number satisfying $z^{2} + \frac{3}{z ^{2}} = - 4$ , then the sum of the absolute values of the imaginary parts of the roots of the equation is

$0$

$23$

$2$

$23 + 2$

$z^{4} + 4 z^{2} + 3 = 0$
$(z^{2} + 1) (z^{2} + 3) = 0$
$\Rightarrow z^{2} = - 1$ or $- 3$
$\Rightarrow z = \pm i, \pm 3 i$
Hence, the required sum is $∣ 1 ∣ + ∣ - 1 ∣ + ∣ ∣ 3 ∣ ∣ + ∣ ∣ - 3 ∣ ∣$

Q. If z is a complex number satisfying z2+z23​=−4 , then the sum of the absolute values of the imaginary parts of the roots of the equation is

0

23​

2

23​+2

z4+4z2+3=0 (z2+1)(z2+3)=0 ⇒z2=−1 or −3 ⇒z=±i,±3​i Hence, the required sum is ∣1∣+∣−1∣+∣∣​3​∣∣​+∣∣​−3​∣∣​

Q. If $z$ is a complex number satisfying $z^{2} + \frac{3}{z ^{2}} = - 4$ , then the sum of the absolute values of the imaginary parts of the roots of the equation is

$0$

$23$

$2$

$23 + 2$

$z^{4} + 4 z^{2} + 3 = 0$
$(z^{2} + 1) (z^{2} + 3) = 0$
$\Rightarrow z^{2} = - 1$ or $- 3$
$\Rightarrow z = \pm i, \pm 3 i$
Hence, the required sum is $∣ 1 ∣ + ∣ - 1 ∣ + ∣ ∣ 3 ∣ ∣ + ∣ ∣ - 3 ∣ ∣$