question_answer

Let \[z\in C\]be such that \[\left| z \right|<1.\]If \[\omega =\frac{5+3z}{5(1-z)},\] then:-             [JEE Main 9-4-2019 Afternoon]

A) \[5\operatorname{Im}(\omega )<1\]             

B) \[4\operatorname{Im}(\omega )>5\]

C) \[5Re(\omega )>1\]    

D) \[5Re(\omega )>4\]

Correct Answer: C

Solution :

\[\left| z \right|<1\]           \[5\omega (1-z)=5+3z\]           \[5\omega -5\omega z=5+3z\]           \[z=\frac{5\omega -5}{3+5\omega }\]           \[|z|=5\left| \frac{\omega -1}{3+5\omega } \right|<1\]           \[5|\omega -1|<|3+5\omega |\] \[5|\omega -1|<5\left| \omega +\frac{3}{5} \right|\] \[|\omega -1|<5\left| \omega -\left( -\frac{3}{5} \right) \right|\]