If $z = a + i b$ satisfy $(4 + 2 i) z + (8 - 2 i) z^{'} = - 2 + 10 i,$ where $z^{'}$ is the complex conjugate of $z$ , then the value of $| a + 2 b |$ is

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Solution

Given : $(4 + 2 i) z + (8 - 2 i) z^{'} = - 2 + 10 i$
$z = a + i b, z^{'} = a - i b$
Now,
$(4 + 2 i) z + (8 - 2 i) z^{'} = - 2 + 10 i \Rightarrow 4 (z + 2 z^{'}) + 2 i (z - z^{'}) = - 2 + 10 i \Rightarrow 4 (3 a - i b) - 4 b = - 2 + 10 i \Rightarrow (12 a - 4 b) - i (4 b) = - 2 + 10 i \Rightarrow 12 a - 4 b = - 2, - 4 b = 10 \Rightarrow b = - \frac{5}{2}, a = - 1 ∴ | a + 2 b | = 6$

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