Question

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

Answer 1

Hint: In this question, we need to find the minimum value of $$\left|Z\right|+\left|Z-1\right|$$ 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 $$\left|-Z\right|=\left|Z\right|$$ and $$\left|Z_1+Z_2\right|⩽\left|Z_1\right|+\left|Z_2\right|$$

Complete step-by-step answer:
We have been given that $$Z$$ is any complex number.
Also, we know that $$\left|Z\right|+\left|Z-1\right|$$
According to the property of a complex number, we get
$$\left|-Z\right|=\left|Z\right|$$
Thus, we can say that
$$\left|Z-1\right|=\left|1-Z\right|$$ ….. (1)
By applying the properties of a complex number and from equation (1) , we get
$$\left|Z\right|+\left|\text{1 }-\text{ Z}\right|=\left|Z\right|+\left|1-Z\right|⩾\left|Z+\left(1-Z\right)\right|=1$$
So, we get
$$\left|Z\right|+\left|Z-1\right|⩾1$$
Hence, the minimum value of $$\left|Z\right|+\left|Z-1\right|$$ 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 $$\left|Z_1+Z_2\right|⩽\left|Z_1\right|+\left|Z_2\right|$$ 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.