Question
If $$f\left( z \right) = \frac{{7 - z}}{{1 - {z^2}}} ,$$ where $$z = 1 + 2i,$$ then $$\left| {f\left( z \right)} \right|$$ is equal to :
A.
$$\frac{{\left| z \right|}}{2}$$
B.
$${\left| z \right|}$$
C.
$$2{\left| z \right|}$$
D.
None of these
Answer :
$$\frac{{\left| z \right|}}{2}$$
Solution :
$$\eqalign{
& z = 1 + 2i \cr
& \Rightarrow \left| z \right| = \sqrt {1 + 4} = \sqrt 5 \cr
& \therefore \,f\left( z \right) = \frac{{7 - z}}{{1 - {z^2}}} = \frac{{7 - 1 - 2i}}{{1 - {{\left( {1 + 2i} \right)}^2}}} \cr
& = \,\frac{{6 - 2i}}{{1 - \left( {1 - 4 + 4i} \right)}} = \frac{{6 - 2i}}{{4 - 4i}} = \frac{{3 - i}}{{2 - 2i}} \cr
& \Rightarrow \,\left| {f\left( z \right)} \right| = \left| {\frac{{3 - i}}{{2 - 2i}}} \right| = \frac{{\left| {3 - i} \right|}}{{\left| {2 - 2i} \right|}} \cr
& = \,\frac{{\sqrt {9 + 1} }}{{\sqrt {4 + 4} }} = \frac{{\sqrt 5 }}{2} = \frac{{\left| z \right|}}{2} \cr} $$