Question

If $z_1$ and $z_2$ are complex numbers such that $Re\left(z_1\right)=|z_1-2|,Re\left(z_2\right)=|z_2-2|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then the value of $Im(z_1+z_2)$ is

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{4}{\sqrt{3}}$
• C. $2\sqrt{3}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $\frac{4}{\sqrt{3}}$

Let $z_1=x_1+i{y}_1$and $z_2=x_2+i{y}_2$
Then, $x_1^2=(x_1-2{)}^2+y_1^2$ and
$x_2^2=(x_2-2{)}^2+y_2^2$
$\Rightarrow 4x_1=y_1^2+4$ and $4x_2=y_2^2+4$
On subtraction, we get
$4(x_1-x_2)=y_1^2-y_2^2=(y_1+y_2)(y_1-y_2)$
Hence, $y_1+y_2=\frac{4(x_1-x_2)}{y_1-y_2}\cdots \left(1\right)$

Also, $\arg (z_1-z_2)=\frac{\pi }{3}$
$\therefore \tan \frac{\pi }{3}=\frac{y_1-y_2}{x_1-x_2}$
$\Rightarrow \sqrt{3}=\frac{y_1-y_2}{x_1-x_2}\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$ we have
$Im(z_1+z_2)=y_1+y_2=\frac{4}{\sqrt{3}}$