The complex number `z` satisfies `z+|z|=2+8i`. find the value of `|z|-8`

Answer

17

Explanation

Let `z = a+ ib`
`rArr |z|^(2) = a^(2) + b^(2)`
Now `z +|z| = 2 + 8i`
`rArr a+ib+sqrt(a^(2) +b^(2)) = 2+8i`
`rArr a+ sqrt(a^(2) + b^(2)) = 2, b = 8`
`rArr a+sqrt(a^(2)+64)=2`
`rArr a^(2) + 64 =(2-a)^(2) = a^(2) -4a+4`,
`rArr 4a = - 60, a = - 15`
Thus, `a^(2) + b^(2) = 225 + 64 = 289`
`therefore |z| = sqrt(a^(2)+b^(2)) = sqrt(289) = 17`