Animated Solution for Mathematics - Complex Numbers: If $z_1,z_2$ are complex numbers such that $\text{Re}(z_1)=|z_1-1|$, $\text{Re}(z_2)=|z_2-1|$, and $\text{arg}(z_1-z_2)=\frac{\pi }{6}$, then $\text{Im}(z_1+z_2)$ is equal to

We are given two complex numbers, $z_1$ and $z_2$, satisfying the condition $\text{Re}(z)=|z-1|$. Let $z=x+iy$. The real part is $x$, and the modulus $|z-1|$ represents the distance from $z$ to the point $(1,0)$ in the complex plane.

The constraint can be written as:

Squaring both sides yields $x^2=(x-1{)}^2+y^2$. Expanding the right side, we obtain:

The $x^2$ terms cancel out, leaving us with the equation of a parabola:

The points $z_1$ and $z_2$ lie on this parabola. We are given that $\text{arg}(z_1-z_2)=\frac{\pi }{6}$.

The argument of the difference of two complex numbers represents the angle of the vector connecting them. Consequently, the slope $m$ of the chord joining $z_1$ and $z_2$ is:

For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on the parabola $y^2=2x-1$, we can express $x$ as $x=\frac{y^2+1}{2}$. The slope of the chord is given by:

Simplifying this expression, we get:

Equating this to our known slope of $\frac{1}{\sqrt{3}}$, we have:

Solving for the sum of the imaginary parts, we find:

Since the imaginary part of $z_1+z_2$ is defined as $y_1+y_2$, the final result is $2\sqrt{3}$.