If z is a complex number lying in the fourth quadrant of Argand plane and ∣∣∣[kzk+1]+2i∣∣∣>√2 for all real values of k(k≠−1), then range of arg(z) is
A
(−π8,0)
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B
(−π6,0)
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C
(−π4,0)
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D
None of these
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Solution
The correct option is C(−π4,0) z1=kz(k+1) represents any point lying on the line joining origin and z.
Given, ∣∣∣[kzk+1]+2i∣∣∣>√2
Hence, kzk+1 will lie outside the circle |z1+2i|=√2.
So, z should lie in the shaded region.
Now △OPA is a rightangled triangle OP2=OA2−AP2=4−2=2⇒OP=AP⇒∠POA=π4 ∴−π4<arg(z)<0