Answer:
Option D
Explanation:
Given ,
|\frac{z+3i}{3z+i}| <1
|\frac{x+(y+3)i}{3(x+iy)+i}| <1 [Let z=x+iy]
| x+(y+3)i| < | 3x+(3y+1)i|
\Rightarrow x^{2} +(y+3)^{2} < 9 x^{2}+(3y+1)^{2}
\Rightarrow x^{2}+y^{2}+6y+9 < 9 x^{2}+9 y^{2}+6y+1
\Rightarrow 8x^{2}+8 y^{2}-8 >0
\Rightarrow x^{2}+y^{2}-1 >0
\frac{k-1}{k}, \frac{k-2}{k} lie on locus of z
\therefore \frac{(k-1)^{2}}{k^{2}}+\frac{(k-2)^{2}}{k^{2}}-1 >0
\Rightarrow k^{2}-2k+1+k^{2}-4k+4-k^{2} >0
\Rightarrow k^{2}-6k+5 >0 \Rightarrow (k-5)(k-1) > 0
\Rightarrow k ε (-\infty ,1) \cup (5, \infty)