Question

If \(z\) is a complex number, then the minimum value of \(|z-2|+|z-4|\) is

• A. \(\sqrt{20}\)
• B. \(4\)
• C. \(6\)
• D. \(\sqrt{6}\)
• E. \(2\)

Hint:

For expressions like \(|z-a|+|z-b|\), think of distances in the complex plane. The minimum value is the straight-line distance between the fixed points \(a\) and \(b\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem has a geometric interpretation in the complex plane. The expression \(|z - z_1|\) represents the distance between the point representing the complex number \(z\) and the point representing the complex number \(z_1\). The expression \(|z-2| + |z-4|\) represents the sum of the distances from a point \(z\) to two fixed points, 2 and 4, on the real axis.
Step 2: Key Formula or Approach:
We can use the triangle inequality. For any two complex numbers \(z_1\) and \(z_2\), the inequality states \(|z_1| + |z_2| \ge |z_1 + z_2|\).
Let's rewrite the expression to apply this. Let \(z_1 = z-2\) and \(z_2 = 4-z\).
Then, the expression is \(|z-2| + |4-z|\).
According to the triangle inequality:
\[ |z-2| + |4-z| \ge |(z-2) + (4-z)| \] Step 3: Detailed Explanation:
Applying the inequality from Step 2:
\[ |z-2| + |z-4| = |z-2| + |4-z| \ge |(z-2) + (4-z)| \] \[ \ge |z - 2 + 4 - z| \] \[ \ge |2| = 2 \] So, the minimum possible value for the sum is 2.
Geometrically, let P be the point for \(z\), A be the point for 2, and B be the point for 4. The expression represents the sum of distances PA + PB. The minimum value of this sum occurs when the point P lies on the line segment connecting A and B. In this case, the sum of the distances is simply the distance between A and B.
Distance between A(2) and B(4) is \(|4 - 2| = 2\).
Step 4: Final Answer:
The minimum value of the expression is 2. Therefore, option (E) is the correct answer.