Let Z_1 and Z_2 be two complex numbers such that (Z_1-Z_2)=4, and Z_1,Z_2 satisfy |Z-3|=Re…

Question

Let $Z_1$ and $Z_2$ be two complex numbers such that $\arg(Z_1-Z_2)=\dfrac\pi4$, and $Z_1,Z_2$ satisfy $|Z-3|=\operatorname{Re}(Z)$. Then the imaginary part of $Z_1+Z_2$ is equal to

Answer: 6

Accepted Answer

Let $Z_1=a_1+ib_1$, $Z_2=a_2+ib_2$. From $\arg(Z_1-Z_2)=\dfrac\pi4$: $b_1-b_2=a_1-a_2$.

From $|Z_i-3|=\operatorname{Re}(Z_i)$: $(a_i-3)^2+b_i^2=a_i^2$, i.e. $b_i^2=6a_i-9$. Subtracting for $i=1,2$ and using $b_1-b_2=a_1-a_2$ gives $b_1+b_2=6$.

Solutions · 1