Let z_1 and z_2 be any two non-zero complex numbers such that 3|z_1|=4|z_2|.

Question

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1|=4|z_2|$. If $z=\dfrac{3z_1}{2z_2}+\dfrac{2z_2}{3z_1}$, then:

Accepted Answer

Writing $z_1=r_1e^{i\theta}$, $z_2=r_2e^{i\varphi}$ with $3r_1=4r_2$:

$$z=\frac32\cos(\theta-\varphi)+i\left(2-\frac12\right)\sin(\theta-\varphi)\cdots$$

leads to $\dfrac32\le|z|\le\dfrac52$, which does not match any single stated option exactly, so the answer is (none of the above).

Solutions · 1