$θ \in [0, 2 π] and z_{1}, z_{2}, z_{3} are three complex numbers such that they are collinear and$ $(1 + | \sin θ |) z_{1} + (| \cos θ | - 1) z_{2} - \sqrt{2} z_{3} = 0 . If at least one of the complex numbers z_{1}, z_{2}, z_{3}$ $is non- zero then number of possible values of θ is$

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answer is D.

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Detailed Solution

$I f$ $z_{1}, z_{2}, z_{3}$ $a r e c o l l i n e a r a n d$ $a z_{1} + b z_{2} + c z_{3} = 0$ $t h e n$ . $a + b + c = 0$

$H e n c e$ $1 + |\sin θ| + |\cos θ| - 1 - \sqrt{2} = 0 \Rightarrow |\sin θ| + |\cos θ| = \sqrt{2}$

$\Rightarrow θ \in \{\frac{π}{4}, \frac{3π}{4}, \frac{5π}{4}, \frac{7π}{4}\}$

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