Question

$z$ is a complex number satisfying $|z-2-3i|=\alpha$ 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 $|z-2-3i|=\alpha$, 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 $(2+3i)\&$ 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 $(2+3i)$ i.e $\alpha \sqrt{3}$