f  z1, z2𝑧1, 𝑧2 are complex numbers such that Re(z1)=|z1−2|Re𝑧1=𝑧1-2 , Re(z2)=|z2−2|Re𝑧2=𝑧2-2 and arg(z1−z2)=π3,arg𝑧1-𝑧2=𝜋3, then Im(z1+z2)=

The given conditions are:

1. Re(z1) = |z1 - 2|
2. Re(z2) = |z2 - 2|
3. arg(z1 - z2) = π/3

From the first two conditions, we can infer that the real parts of z1 and z2 are non-negative (since the absolute value of a complex number is always non-negative). This means that z1 and z2 lie in the first or fourth quadrant of the complex plane.

The third condition tells us that the argument of the complex number (z1 - z2) is π/3. This means that the angle that the line segment joining z1 and z2 makes with the positive real axis is π/3.

Since z1 and z2 lie in the first or fourth quadrant, the only way for the angle between them to be π/3 is if one of them is on the positive real axis and the other is in the first quadrant. Without loss of generality, let's assume that z1 is on the positive real axis and z2 is in the first quadrant.

Then, z1 = x + 0i (for some real number x) and z2 = a + bi (for some real numbers a and b, with b > 0).

The condition Re(z1) = |z1 - 2| becomes x = |x - 2|. This has two solutions: x = 1 or x = 2. Since z1 is on the positive real axis, we must have x = 2.

The condition Re(z2) = |z2 - 2| becomes a = |a + bi - 2| = sqrt((a - 2)^2 + b^2). Squaring both sides gives a^2 = (a - 2)^2 + b^2. Simplifying this gives a = 1 + b^2/2.

The condition arg(z1 - z2) = π/3 becomes arg(2 - (1 + b^2/2 + bi)) = π/3. This simplifies to arg(1 - b^2/2 - bi) = π/3. Since the argument of a complex number is the angle it makes with the positive real axis, this means that the angle between the positive real axis and the line segment from 0 to 1 - b^2/2 - bi is π/3. This gives us another equation for b: tan(π/3) = b/(1 - b^2/2). Solving this for b gives b = sqrt(3).

Substituting b = sqrt(3) into the equation a = 1 + b^2/2 gives a = 2.

So, z1 = 2 + 0i and z2 = 2 + sqrt(3)i.

Then, Im(z1 + z2) = Im(2 + 0i + 2 + sqrt(3)i) = sqrt(3).