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 One root lies inside the unit circle and one outside if

• A. $-1<\lambda <1$
• B. $\lambda >1$
• C. $\lambda <1$
• D. None of these

Answer 1

Answer: (B)

$\lambda >1$

Given:

$z^2+2z\lambda +1=0$

$\Rightarrow z^2+2z\lambda +{\lambda }^2={\lambda }^2-1$

$\Rightarrow (z+\lambda {)}^2=({\lambda }^2-1)$

For the root to lie inside and outside of the unit circle:

$({\lambda }^2-1)\ge 0$

$\Rightarrow {\lambda }^2\ge 1$

$\therefore \lambda >1$