logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
0 votes

If for any complex number $z$, $|z-1|=1$, then $\large\frac{z-2}{z}=?$

$\begin{array}{1 1}(A)\tan (arg \;z) \\ (B) \cot (arg\; z) \\ (C) i \tan (arg\;z) \\(D) i \cot (arg \;z) \end{array}$

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • $cos\theta-1=-2sin^2\large\frac{\theta}{2}$
  • $cos\theta+1=2cos^2\large\frac{\theta}{2}$
  • $sin\theta=2sin\large\frac{\theta}{2}$$cos\large\frac{\theta}{2}$
Given: $|z-1|=1$
Let $ z-1=cos\;\theta+isin\;\theta$
$\Rightarrow\:z-2=-1+cos\;\theta+isin\;\theta$
$=2sin\large\frac{\theta}{2}$$(-sin\large\frac{\theta}{2}$$+icos\large\frac{\theta}{2}$$)$
$=2isin\large\frac{\theta}{2}$$(cos\large\frac{\theta}{2}$$+isin\large\frac{\theta}{2}$$)$
and
$z=1+cos\theta+isin\theta$
$=2cos\large\frac{\theta}{2}$$(cos\large\frac{\theta}{2}$$+isin\large\frac{\theta}{2}$$)$
$\Rightarrow\:arg(z)=\large\frac{\theta}{2}$
$\therefore\:\large\frac{z-1}{z}$$=itan\large\frac{\theta}{2}$
$=itan(arg\:z)$
answered Aug 6, 2013 by rvidyagovindarajan_1
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...