Question

Let $z$ be a complex number satisfying $z^2+2z\lambda +1=0$, where $\lambda$ is a parameter which can take any real value. For very large value of $\lambda$, the roots are approximately

• A. $-2\lambda ,\frac{1}{\lambda }$
• B. $-\lambda ,\frac{-1}{\lambda }$
• C. $-2\lambda ,-\frac{1}{2\lambda }$
• D. None of these

Answer 1

Answer: (C)

$-2\lambda ,-\frac{1}{2\lambda }$

$z=-\lambda \pm \sqrt{{\lambda }^2-1}$
When $\lambda$ is very large, then
$z=-\lambda -\sqrt{{\lambda }^2-1}\approx -2\lambda$
$z=-\lambda +\sqrt{{\lambda }^2-1}=\frac{(-\lambda +\sqrt{{\lambda }^2-1})(-\lambda -\sqrt{{\lambda }^2-1})}{(-\lambda -\sqrt{{\lambda }^2-1})}$
$=\frac{1}{-\lambda -\sqrt{{\lambda }^2-1}}=-\frac{1}{2\lambda }$