Question

For a complex number z, the minimum value of $\left| z \right|+\left| z-4 \right|$ is:
(a) Zero
(b) 2
(c) 4
(d) 8

Answer 1

Hint: We have to find the minimum value of $\left| z \right|+\left| z-4 \right|$ where z is a complex number. We will find the minimum value by using the triangle inequality which says that $\left| a \right|+\left| b \right|\ge \left| a-b \right|$. Now, substitute “a” as “z” and “b” as “z – 4” then the left hand side of this inequality will be equal to $\left| z \right|+\left| z-4 \right|$ and substitute these values of “a and b” in the right hand side of the inequality also. The minimum value occurs when there is an equal sign then $\left| a-b \right|$ is the answer.

Complete step by step solution:
We have given that:
$\left| z \right|+\left| z-4 \right|$
We have to find the minimum value of the above relation. In the above relation, z is a complex number.
We are going to find the minimum value of this expression by using the triangle inequality which states that:
$\left| a \right|+\left| b \right|\ge \left| a-b \right|$
In the above inequality, “a” and “b” are complex numbers. Now, substituting z in place of “a” and “z – 4” in place of “b” in the above inequality we get,
$\begin{align}
  & \left| z \right|+\left| z-4 \right|\ge \left| z-\left( z-4 \right) \right| \\
 & \Rightarrow \left| z \right|+\left| z-4 \right|\ge \left| 4 \right| \\
\end{align}$
As you can see that left hand side of the above relation is the expression of which we have to find the minimum value so minimum value of the above relation exists when in the above inequality, equality occurs meaning:
$\left| z \right|+\left| z-4 \right|=\left| 4 \right|$
Now, the value of $\left| 4 \right|$ is equal to:
$\begin{align}
  & \sqrt{{{\left( 4 \right)}^{2}}} \\
 & =4 \\
\end{align}$
From the above solution, the minimum value of $\left| z \right|+\left| z-4 \right|$ is equal to 4.
Hence, the correct option is (c).

Note: The point where you make a mistake in the question is while taking the square root of ${{\left( 4 \right)}^{2}}$ in the hastiness of solving a problem you might have written the square root as 2. This is a very silly mistake that I have seen often. And one more thing you might have thought while taking the square root of ${{\left( 4 \right)}^{2}}$ why we have only considered positive value i.e. 4 and ignore the negative value i.e. – 4. The reason is 4 is the modulus of the complex number 4 and modulus can take only positive value.