Let $z$ be a complex number such that $|z+2|=1$ and $\text{Im}\left(\dfrac{z+1}{z+2}\right)=\dfrac{1}{5}$. Then the value of $|\text{Re}(z+2)|$ is:
Step-by-Step Solution
Key Concept: Let $z+2=\cos\theta+i\sin\theta$ (unit circle). $\frac{z+1}{z+2}=1-\frac{1}{z+2}=1-(\cos\theta-i\sin\theta)=(1-\cos\theta)+i\sin\theta$. $\text{Im}=\sin\theta=1/5$.
$\sin\theta=1/5$, $|\cos\theta|=\frac{2\sqrt{6}}{5}$.
Correct Answer: 1