Question

Let $z=a+ib$ (where $a,b\in R$ and $i=\sqrt{-1})$ such that $|2z+3i|=|z^2|.$ Identify the correct statement(s)?

• A. $|z{|}_{\text{maximum}}$ is equal to $3$.
• B. $|z{|}_{\text{minimum}}$ is equal to $1$.
• C. If $|z|\text{is maximum}$, then $a^3+b^3$ is equal to $27$.
• D. $|z|\text{is minimum}$, then $(a^2+2b^2)$ is $2$.

Answer 1

Answer: (A), (B), (C), (D)

$|z|\text{is minimum}$, then $(a^2+2b^2)$ is $2$.

$|2z+3i|\le 2|z|+3$

$|z{|}^2\le 2|z|+3$

$\Rightarrow 0\le |z|\le 3----\left(1\right)$

$|2z+3i|\ge |2|z|-3|$

$|z^2|\ge |2|z|-3|\Rightarrow |z|\ge 1----\left(2\right)$

So, $\left(1\right)$ and $\left(2\right)$ gives

$1\le |z|\le 3$

$|z{|}_{\text{maximum}}\Rightarrow z=3i$

So, $a=0,b=3$ &

$|z{|}_{\text{minimum}}\Rightarrow z=-i$

So, $a=0,b=-1$