Question

The minimum value of |z+1|+|z-2| is equal to

• A. 1
• B. 2
• C. 3
• D. 4
• E. 0

Answer 1

The Correct Option is C

To minimize \( |z+1| + |z-2| \), we need to find the minimum sum of the distances from \( z \) to the points \( -1 \) and \( 2 \) on the complex plane.

This is a classical problem known as the triangle inequality. The minimum value occurs when \( z \) lies on the line segment joining \( -1 \) and \( 2 \). The distance between \( -1 \) and \( 2 \) is \( |2 - (-1)| = 3 \).
Thus, the minimum value of \( |z+1| + |z-2| \) is \( 3 \), which occurs when \( z \) lies on the segment between \( -1 \) and \( 2 \).

The correct option is (C) : 3

Answer 2

Let z be a complex number. We want to find the minimum value of |z+1| + |z-2|.

Geometrically, |z+1| represents the distance between the point z in the complex plane and the point -1, and |z-2| represents the distance between the point z and the point 2.

Therefore, we want to find the point z in the complex plane that minimizes the sum of its distances to -1 and 2. The shortest distance between two points is a straight line. Thus, the minimum value occurs when z lies on the real axis between -1 and 2.

In this case, the minimum value is simply the distance between -1 and 2:

Minimum value = |2 - (-1)| = |2 + 1| = 3

Alternatively, we can use the triangle inequality: |z+1| + |z-2| = |z+1| + |-(2-z)| = |z+1| + |2-z| ≥ |(z+1) + (2-z)| = |3| = 3

Equality holds when (z+1) and (2-z) have the same argument. This occurs when z is a real number between -1 and 2.

Therefore, the minimum value of |z+1| + |z-2| is equal to 3.