then which of the following is correct?
A) \[\arg \,(z-1)=2argz\]
B) \[2\arg \,(z)=\frac{2}{3}arg({{z}^{2}}-z)\]
C) \[\arg (z-1)=\arg (z+1)\]
D) \[\arg z=2\arg (z+1)\]
Correct Answer: A
Solution :
Since \[\left| z-1 \right|= 1\,\,\,\, \therefore z -1 = {{e}^{i\theta }}\], where \[\arg \,\,\left| z-1 \right|=\theta \] \[\therefore \,\,z=1+cos\theta \,\,+\,\,i\,\sin \theta \] \[=\,\,\,2\cos \frac{\theta }{2}\left[ \cos \frac{\theta }{2}+i\sin \frac{\theta }{2} \right]\] \[=\,\,2\cos \frac{\theta }{2}.\,{{e}^{i\theta /2}}=2{{\cos }^{2}}\frac{\theta }{2}+2i\,\sin \frac{\theta }{2}\cos \frac{\theta }{2}\] Thus, \[\arg \left( z -1 \right) = 2 arg z.\]
You need to login to perform this action.
You will be redirected in
3 sec
