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Consider the locus of the complex number z in the argand plane given by $R e (z) - 2 = | z - 7 + 2 i | .$ Let $P (z_{1})$ and $Q (z_{2})$ be two complex numbers satisfying the given locus and also satisfying $ar g (\frac{z_{1} - (2 + α i)}{z_{2} - (2 + α i)}) = \frac{π}{2} (α \in R) .$ The minimum value of PQ is
[Note: $R e (z)$ denotes real part of complex number z and $i^{2} = - 1$ .]

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Detailed Solution

Given, $R e (z) - 2 = | z - 7 + 2 i |$

$\Rightarrow {(x - 2)}^{2} = {(x - 7)}^{2} + {(y - 2)}^{2}$

$\Rightarrow {(y - 2)}^{2} = 10 (x - \frac{9}{2})$

So, the given locus is that of a parabola with directrix
x = 2 and focus (7, 2).
Clearly, minimum PQ = l(L.R.) = 10

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