Let z₁ and z₂ are non-zero (given) complex numbers and k be any positive real number. Consider the system of equations $|3 z - z_{1} - 2 z_{2}| = |z_{1} - z_{2}| and \arg (\frac{z_{1} - z_{2}}{z - {kz}_{1} - (1 - k) z_{2}}) = \pm \frac{π}{2}$ . Then

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

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 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 $Arg (\frac{z_{1} - z_{2}}{z - ({kz}_{1} + (1 - k) z_{2})}) = \pm \frac{π}{2}$ .
Angle between the line segment joining $z_{1} and z_{2}; z$ $and {kz}_{1} + (1 - k) z_{2} is \frac{π}{2} \frac{z_{1} + 2 z_{2}}{3}$ is a point on segment AB such that AC : CS = 2:1 and D $({kz}_{1} + (1 - k) z_{2})$

is a point on segment AB such that AD:DB = 1 - k : k
$\Rightarrow BD = k |(z_{1} - z_{2})|$
(a) for nosol BD > BB₁
$\Rightarrow k |z_{1} - z_{2}| > \frac{2}{3} |z_{1} - z_{2}| \Rightarrow k > \frac{2}{3}$
(b) for more than one sol. 0 < BD < BB₁
$\Rightarrow 0 < k |z_{1} - z_{2}| < \frac{2}{3} |z_{1} - z_{2}| 0 < k < 2 / 3 .$