Question

For any complex number \[Z\], the minimum value of \[\left| Z \right|{\text{ }} + {\text{ }}\left| {Z{\text{ }} - {\text{ }}1} \right|\] is
A. 0
B. 1
C. 2
D. -1

Answer 1

Hint: In this question, we need to find the minimum value of \[\left| Z \right|{\text{ }} + {\text{ }}\left| {Z{\text{ }} - {\text{ }}1} \right|\] if \[Z\] is a complex number. For this, we have to use the following property of a complex number to get the required minimum value.

Formula used: We will use the following property of a complex number to solve this problem.
If \[Z\] is an any complex number, then \[\left| { - Z} \right|{\text{ }} = {\text{ }}\left| Z \right|\] and \[\left| {{Z_1} + {Z_2}} \right|{\text{ }} \leqslant {\text{ }}\left| {{Z_1}} \right| + \left| {{Z_2}} \right|\]

Complete step-by-step answer:
We have been given that \[Z\] is any complex number.
Also, we know that \[\left| Z \right|{\text{ }} + {\text{ }}\left| {Z{\text{ }} - {\text{ }}1} \right|\]
According to the property of a complex number, we get
\[\left| { - Z} \right|{\text{ }} = \left| Z \right|\]
Thus, we can say that
\[\left| {Z{\text{ }} - {\text{ }}1} \right| = \left| {1 - Z} \right|\] ….. (1)
By applying the properties of a complex number and from equation (1) , we get
\[\left| Z \right|{\text{ }} + {\text{ }}\left| {{\text{1 }} - {\text{ Z}}} \right| = \left| Z \right| + \left| {1 - Z} \right| \geqslant \left| {Z + \left( {1 - Z} \right)} \right| = 1\]
So, we get
\[\left| Z \right|{\text{ }} + {\text{ }}\left| {Z{\text{ }} - {\text{ }}1} \right| \geqslant 1\]
Hence, the minimum value of \[\left| Z \right|{\text{ }} + {\text{ }}\left| {Z{\text{ }} - {\text{ }}1} \right|\] is 1.
Therefore, the correct option is (B).

Additional information: We can say that the modulus of a complex number is defined as the distance from the origin of the point on the argand plane denoting the complex number z. Here, the property of a complex number such as \[\left| {{Z_1} + {Z_2}} \right|{\text{ }} \leqslant {\text{ }}\left| {{Z_1}} \right| + \left| {{Z_2}} \right|\] is known as the triangle’s inequality of complex numbers. It states that, the sum of any two sides of a triangle seems to be greater than or equal to the third side.

Note: Here, students make mistakes in applying the properties of a complex number. It is necessary to apply the correct property at the correct place and we need to do simplification according to it.