If Z_1,Z_2 are complex numbers such that Re(Z_1)=|Z_1-1|, Re(Z_2)=|Z_2-1| and (Z_1-Z_2)=6,…

Question

If $Z_1,Z_2$ are complex numbers such that $\operatorname{Re}(Z_1)=|Z_1-1|$, $\operatorname{Re}(Z_2)=|Z_2-1|$ and $\arg(Z_1-Z_2)=\dfrac\pi6$, then $\operatorname{Im}(Z_1+Z_2)$ is equal to:

Accepted Answer

For each $i$: $y_i^2=2x_i-1$. Subtracting gives $y_1^2-y_2^2=2(x_1-x_2)$, so $y_1+y_2=\dfrac{2(x_1-x_2)}{y_1-y_2}$.

Using $\dfrac{y_1-y_2}{x_1-x_2}=	an\dfrac\pi6=\dfrac1{\sqrt3}$: $y_1+y_2=2\sqrt3.$

Solutions · 1