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: Interpret the expression geometrically.
A complex number \(z\) can be represented as a point in the Argand plane. The quantity \(|z-2|\) is the distance of the point \(z\) from the point \(2\), and \(|z-4|\) is the distance of the point \(z\) from the point \(4\).

Step 2: Identify the fixed points.
The points \(2\) and \(4\) lie on the real axis of the complex plane. So we are looking for the minimum value of:
\[ |z-2|+|z-4| \] which is the sum of distances of a moving point \(z\) from the two fixed points \(2\) and \(4\).

Step 3: Apply the triangle inequality idea.
For any point \(z\), the sum of distances from \(z\) to two fixed points is always at least the direct distance between those two fixed points. Therefore,
\[ |z-2|+|z-4|\geq |(4)-(2)| \] \[ =|2|=2 \]

Step 4: Compute the distance between the fixed points.
Since both \(2\) and \(4\) are real numbers, their distance is simply:
\[ |4-2|=2 \] So the minimum possible value cannot be less than \(2\).

Step 5: Check whether this minimum is actually attainable.
Equality in the triangle inequality is possible when the point \(z\) lies on the line segment joining \(2\) and \(4\). Since these points lie on the real axis, any real number between \(2\) and \(4\) will work.

Step 6: Verify with a specific value.
Take \(z=3\). Then:
\[ |z-2|+|z-4|=|3-2|+|3-4|=1+1=2 \] So the lower bound \(2\) is actually achieved.

Step 7: State the final answer.
Hence, the minimum value of \(|z-2|+|z-4|\) is:
\[ 2 \] This matches option \((5)\).