Question

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

• A. ${\left|z\right|}_{maximum}isequalto3$.
• B. ${\left|z\right|}_{minimum}isequalto1$.
• C. $If{\left|z\right|}_{maximum}$ $when$ $z=\alpha +i\beta$ $(\alpha ,\beta \in R$ $and$ $i=\sqrt{-1})$ $then$ $({\alpha }^3+{\beta }^3)$ $isequalto27$.
• D. $If{\left|z\right|}_{minimum}$ $when$ $z=x+iy(x,y\in R)$ $and$ $i=\sqrt{-1})$ $then$ $(x^2+{2y}^2)$ $isequalto2$.

Answer 1

The correct option is B ${\left|z\right|}_{minimum}isequalto1$.

$|2z+3i|=|z^2|$

$4z^2-9z-12!=z^2$

$3z^2-9z-12!=0$

$z^2-3z-4=0$

$z^2-4z+z-4=0$

$(z-4)(z+1)=0$

so $|z{|}_{min}=1$