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the complex number z satisfies z+|z|=2+8i

then the value of |z| - 8=

 

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\begin{array}{l} \text { Let } z=a+i b \\ \Rightarrow|z|^{2}=a^{2}+b^{2} \\ \text { Now } z+|z|=2+8 i \\ \Rightarrow a+i b+\sqrt{a^{2}+b^{2}}=2+8 i \\ \Rightarrow a+\sqrt{a^{2}+b^{2}}=2, b=8 \\ \Rightarrow a+\sqrt{a^{2}+64}=2 \\ \Rightarrow a^{2}+64=(2-a)^{2}=a^{2}-4 a+4 \\ \Rightarrow 4 a=-60, a=-15 \\ \text { Thus, } a^{2}+b^{2}=225+64=289 \\ \therefore|z|=\sqrt{a^{2}+b^{2}}=\sqrt{289}=17 \end{array}

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