If $z_1$,$z_2$ are two non zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$ then the difference of the amplitude of the ratio of $z_1$ and $z_2$ equal to

Let $z_1=a\left(\cos A+i\sin A\right)\&z_2=b\left(\cos B+i\sin B\right)$

$$\begin{gathered} \left|z_2{+z}_1\right|=\left|z_1\right|+\left|z_2\right| \\ \Rightarrow \left|a\cos A+b\cos B+i\left(a\sin A+b\sin B\right)\right|=a+b \\ \Rightarrow {\left(a\cos A+b\cos B\right)}^2+{\left(a\sin A+b\sin B\right)}^2=a^2{+b}^2+2ab \\ \Rightarrow a^2{+b}^2+2ab\left(\cos A\cos B+\sin A\sin B\right)=a^2{+b}^2+2ab \\ \Rightarrow \cos \left(A-B\right)=1 \\ \therefore A-B=0 \end{gathered}$$