Question

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

• A. $-\pi$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. $0$

Answer 1

Answer: (D)

$0$

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