Question

If z is a complex number lying in the fourth quadrant of Argand plane and $|\left[kz/\right(k+1\left)\right]+2i|>\sqrt{2}$ for all real value of $k(k\ne -1)$, then range of arg (z) is

• A. $(-\frac{\pi }{8},0)$
• B. $(-\frac{\pi }{6},0)$
• C. $(-\frac{\pi }{4},0)$
• D. None of these

Answer 1

The correct option is C $(-\frac{\pi }{4},0)$

$kz/(k+1)$ represents any point lying on the line joining origin and z.
Given,
$\left|\frac{kx}{k+1}+2i\right|>\sqrt{2}$
Hence, $kz/(k+1)$ should lie outside the circle $|z+2i|>\sqrt{2}$. so, z should lie in the shaded region.
$\therefore -\frac{\pi }{4}<arg\left(z\right)<0$