Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to
Step-by-Step Solution
Key Concept: Since $|z|$ is real, equating imaginary parts directly gives $\text{Im}(z)$, reducing the problem to one real equation in $\text{Re}(z)$.
**Step 1: Separate real and imaginary parts**
Let $z=x+iy$. Then $|z|+x+iy=3+i$. Imaginary part: $y=1$. Real part: $|z|+x=3$.
**Step 2: Solve for x**
$\sqrt{x^2+1}=3-x$. Squaring (valid since $x<3$): $x^2+1=9-6x+x^2 \Rightarrow 6x=8 \Rightarrow x=\dfrac{4}{3}$.
**Step 3: Compute |z|**
$|z|=\sqrt{\left(\dfrac{4}{3}\right)^2+1^2}=\sqrt{\dfrac{16+9}{9}}=\dfrac{5}{3}$.
Correct Answer: 2