A relation R on the set of non-zero complex numbers is defined by $z_1{Rz}_2$ if and only if $\frac{z_1-z_2}{z_1+z_2}$ is real. Then $R$ is

Clearly $z_1 R z_1 \text{ as }\frac{z_1-z_1}{z_1+z_1}=0$ is real number. Hence, $R$ is reflexive.

 Also, $R$ is symmetric as if $\frac{z_1-z_2}{z_1+z_2}$ is a real number then $\frac{z_2-z_1}{z_1+z_2}$ is also a real number

$z_1=x_1+i{y}_1$and $z_2=x_2+i{y}_2$ then $z_1 R z_2$ iff $\frac{\left(x_1-x_2\right)+i\left(y_1-y_2\right)}{\left(x_1+x_2\right)+i\left(y_1+y_2\right)}$ is a real number

$\iff \frac{\left(\left(x_1-x_2\right)+i\left(y_1-y_2\right)\right)\left(\left(x_1+x_2\right)-i\left(y_1+y_2\right)\right)}{{\left(x_1+x_2\right)}^2+{\left(y_1+y_2\right)}^2}$ is a real number

$\begin{array}{l}\iff \left(y_1-y_2\right)\left(x_1+x_2\right)-\left(x_1-x_2\right)\left(y_1+y_2\right)=0\\ \iff x_1/y_1=x_2/y_3\end{array}$

Similarly for $z_2=x_2+i{y}_2$ and $z_3=x_3+i{y}_3$ then $z_{2}R{z}_3$ iff $x_2/y_2=x_3/y_3$

If $z_{1}R{z}_2$ and $z_{2}R{z}_3$then $x_1/y_1=x_2/y_2=x_3/y_3$

$\frac{x_1}{y_1}=\frac{x_3}{y_3}\Rightarrow z_{1}R{z}_3$ So, $R$ is transitive

Hence, $R$ is an equivalence relation.