Question

$\theta$ ∈ [0, 2$\pi$] and $z_1,z_2,z_3$ are three complex numbers such that they are collinear and

$(1+|sin\theta |)z_1$ + $(|cos\theta |-1)z_2$ - $\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 $\theta$ is

• A. Infinite
• B. 4
• C. 2
• D. 8

Answer 1

The correct option is B

4

If $z_1,z_2,z_3$ are collinear and $a{z}_1+b{z}_2+c{z}_3$ = 0 then a + b + c = 0. Hence

1 + |sin$\theta$| + |cos$\theta$| - 1 - $\sqrt{2}$ = 0 $\Rightarrow$ |sin$\theta$| + |cos$\theta$| = $\sqrt{2}$

$\Rightarrow$ $\theta$ = $\frac{\pi }{4}$, $\frac{3\pi }{4}$, $\frac{5\pi }{4}$, $\frac{7\pi }{4}$