Question

Column I
Column II : possible argument of z = a + ib

Answer 1

A) For $ab>0$
$\Rightarrow a>0,b>0$ or $a<0,b<0$
$\Rightarrow z$ is $1st$ or $3rd$ quadrant
$\Rightarrow \{\begin{array}{c}\begin{array}{cc}{\tan }^{-1}\frac{b}{a}& a>0,b>0\end{array}\\ \begin{array}{cc}{\tan }^{-1}\frac{b}{a}-\pi & a<0,b<0\end{array}\end{array}$
B) For $ab<0$
$\Rightarrow a<0,b>0$ or $a>0,b<0$
$\Rightarrow z$ in $2nd$ or $4th$ quadrant
$\Rightarrow \{\begin{array}{c}\begin{array}{cc}\pi -{\tan }^{-1}\frac{b}{a}& a<0,b>0\end{array}\\ \begin{array}{cc}-{\tan }^{-1}\frac{b}{a}& a>0,b<0\end{array}\end{array}$
C) $a^2+b^2=0\Rightarrow a=b=0$
$\Rightarrow$ argument is not defined
D) $ab=0\Rightarrow a=0$ or $b=0$
$\Rightarrow \{\begin{array}{c}\begin{array}{cc}{\tan }^{-1}\frac{b}{0}=\frac{\pi }{2}& a=0\end{array}\\ \begin{array}{cc}{\tan }^{-1}\frac{0}{a}=0& b=0\end{array}\end{array}$