If z is a complex number which simultaneously satisfies the equations $3 | z - 12 | = 5 | z - 8 i | and | z - 4 | = | z - 8 |$ , where $i = \sqrt{- 1}$ then Im (z) can be

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Detailed Solution

$\begin{array}{l} ∵ 3 | z - 12 | = 5 | z - 8 i | \\ ∴ 9 | z - 12 |^{2} = 25 | z - 8 i |^{2} \\ \Rightarrow 9 (z - 12) (\bar{z} - 12) = 25 (z - 8 i) (\bar{z} + 8 i) \\ \Rightarrow 9 (z \bar{z} - 12 (z + \bar{z}) + 144) = 25 (z \bar{z} + 8 i (z - \bar{z}) + 64) \\ \Rightarrow 16 z \bar{z} + 108 (z + \bar{z}) + 200 (z - \bar{z}) i + 304 = 0 \\ \Rightarrow 16 (x^{2} + y^{2}) + 216 x - 400 y + 304 = 0 \end{array}$
$\begin{aligned} \Rightarrow 2 (x^{2} + y^{2}) + 27 x - 50 y + 38 = 0 . . . . (i) \\ and | z - 4 | = | z - 8 | \Rightarrow | z - 4 |^{2} = | z - 8 |^{2} \end{aligned} \begin{array}{lrl} \Rightarrow | z |^{2} + 16 - 2 Re (4 z) = | z |^{2} + 64 - 2 Re (8 z) \\ \Rightarrow 8 Re (z) = 48 \\ ∴ Re (z) = 6 \\ \Rightarrow x = 6 (ii) \end{array}$
From Eqs. (i) and (ii) , we get
$\begin{array}{l} 2 (36 + y^{2}) + 162 - 50 y + 38 = 0 \\ \Rightarrow y^{2} - 25 y + 136 = 0 \\ \Rightarrow (y - 17) (y - 8) = 0 \\ \Rightarrow y = 17, 8 \\ ∴ Im (z) = 17, 8 \end{array}$