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

Let $z=a+ib$

As we know

$|z|=\sqrt{a^{2}+b^{2}}$

It is given that

$\\|z|+z=a+\sqrt{a^{2}+b^{2}}+ib\\ |z|+z=3+i\\ a+\sqrt{a^{2}+b^{2}}+ib=3+i\\$

b = 1

$\\a+\sqrt{a^{2}+b^{2}}=3\\ a+\sqrt{a^{2}+1}=3\\ a^{2}+1=(3-a)^{2}\\ a^{2}+1=9-6a+a^{2}\\ 6a=8\\ a=\frac{4}{3}$

$\\|z|=\sqrt{a^{2}+b^{2}}\\ |z|=\sqrt{\left ( \frac{4}{3} \right )^{2}+1^{2}}\\ |z|=\frac{5}{3}$

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