Let $z_{1}$ and $z_{2}$ be non–zero (given) complex numbers and k be any positive real number. Consider the variable complex number z, satisfying the pair of equations $| 3 z - z_{1} - 2 z_{2} | = | z_{1} - z_{2} |$ and $\arg (\frac{z_{1} - z_{2}}{z - k z_{1} - (1 - k) z_{2}}) = \pm \frac{π}{2}$ . Then which of the following is/are TRUE?

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The system of equations has no solution if $k \in (\frac{2}{3}, \infty)$

The system of equations have more than one solutions if $k \in (0, \frac{2}{3})$

The system of equations have no solution if $k \in (0, \frac{2}{3})$

The system of equations have more than one solution if $k \in (\frac{2}{3}, \infty)$

answer is A, B.

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Detailed Solution

$| z - \frac{z_{1} + 2 z_{2}}{3} | = \frac{1}{3} | z_{1} - z_{2} |$ and $A r g (\frac{z_{1} - z_{2}}{z - (k z_{1} + (1 - k) z_{2})}) = \pm \frac{π}{2}$
Angle between the line segment joining $z_{1}$ and $z_{2}$ ; z and $k z_{1} + (1 - k) z_{2}$ is $\frac{π}{2} \frac{z_{1} + 2 z_{2}}{3}$ is a point on segment AB such that AC : CB = 2 : 1and $D (k z_{1} + (1 - k) z_{2})$ is a point on segment AB such that AD : DB = 1 – k : $\Rightarrow B D = k | (z_{1} - z_{2}) |$

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(a) for no sol
$B D > B B_{1} \Rightarrow k | z_{1} - z_{2} | > \frac{2}{3} | z_{1} - z_{2} | \Rightarrow k > \frac{2}{3}$
(b) for more than one sol. $0 < B D < B B_{1}$
$\Rightarrow 0 < k | z_{1} - z_{2} | < \frac{2}{3} | z_{1} - z_{2} |$