Question
If $$z$$ is a complex number such that $$\left| z \right| \geqslant 2,$$ then the minimum value of $$\left| {z + \frac{1}{2}} \right|:$$
A.
is strictly greater than $$\frac{5}{2}$$
B.
is strictly greater than $$\frac{3}{2}$$ but less than $$\frac{5}{2}$$
C.
is equal to $$\frac{5}{2}$$
D.
lie in the interval $$(1, 2)$$
Answer :
is strictly greater than $$\frac{3}{2}$$ but less than $$\frac{5}{2}$$
Solution :
We know minimum value of $$\left| {{Z_1} + {Z_2}} \right|{\text{ is }}\left| {\left| {{Z_1}} \right| - \left| {{Z_2}} \right|} \right|$$
Thus minimum value of $$\left| {Z + \frac{1}{2}} \right|{\text{ is }}\left| {\left| Z \right| - \frac{1}{2}} \right| \leqslant \left| {Z + \frac{1}{2}} \right| \leqslant \left| Z \right| + \frac{1}{2}$$
$$\eqalign{
& {\text{Since, }}\left| Z \right| \geqslant 2{\text{ therefore 2}} - \frac{1}{2} < \left| {Z + \frac{1}{2}} \right| < 2 + \frac{1}{2} \cr
& \Rightarrow \,\,\frac{3}{2} < \left| {Z + \frac{1}{2}} \right| < \frac{5}{2} \cr} $$