Question

$z$ is a complex number satisfying $| z -2-3 i |=\alpha$ and $z _{ m }$ and $z _{ M }$ be the corresponding complex numbers for which $| z +1+ i |$ is minimum and maximum respectively.
Let $z$ is a complex number satisfying $| z -2-3 i |=\alpha$, then the range of $\alpha$ for which maximum value of principal value of $(\arg z)$ can be obtained is

• A. $0<\alpha<1$
• B. $1<\alpha<5$
• C. $\sqrt{5}<\alpha<\sqrt{13}$
• D. $\alpha>\sqrt{13}$

Answer 1

Answer: (D)

Because direction of rays will be in opposite direction as shown in the figure for this $\alpha>5$. Locus of $z$ is a circle having centre at $(2+3 i) \&$ radius $=\alpha$. If circle cuts the real axis at negative side, in that case $\arg z$ will be maximum i.e., $\pi$. This is possible when $\alpha>$ distance between origin and $(2+3 i )$ i.e $\alpha \sqrt{3}$