Let z be a complex number such that |z|+z=3+i (where i=-1).

Question

Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to:

Accepted Answer

$$\left| z \right| + z = 3 + i$$

$$z = 3 - \left| z \right| + i$$

Let  $$3 - \left| z \right| = a \Rightarrow \left| z \right| = \left( {3 - a} \right)$$

$$ \Rightarrow z = a + i \Rightarrow \left| z \right| = \sqrt {{a^2} + 1} $$

$$ \Rightarrow 9 + {a^2} - 6a = {a^2} + 1 \Rightarrow a = {8 \over 6} = {4 \over 3}$$

$$ \Rightarrow \left| z \right| = 3 - {4 \over 3} = {5 \over 3}$$

Solutions · 1