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Checking the answer to complex number question

  • #1
rock.freak667
Homework Helper
6,230
31

Homework Statement


Find an equation connecting x and y for which (z-1)/(z+1) has an argument [itex]\alpha[/itex]


Homework Equations


z=x+iy

arg(z)=tan-1(y/x)


The Attempt at a Solution



[tex]\frac{z-1}{z+1}[/tex]

Substituting z=x+iy

[tex]\Rightarrow \frac{z-1}{z+1}=\frac{(x-1)+iy}{(x+1)+iy}[/tex]

Realizing

[tex]\frac{(x+1)(x-1)+iy(x+1)-iy(x-1)-i^2y^2}{(x+1)^2+y^2}[/tex]

Re:i2=-1

[tex]= \frac{x^2+y^2-1}{(x+1)^2+y^2} +i \frac{2y}{(x+1)^2+y^2}[/tex]

Thus

[tex]tan\alpha = \frac{2y}{x^2+y^2-1}[/tex]

Is this correct? Or should I just put in the form of a circle?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
249
Find an equation connecting x and y for which (z-1)/(z+1) has an argument [itex]\alpha[/itex]

Thus

[tex]tan\alpha = \frac{2y}{x^2+y^2-1}[/tex]

Is this correct? Or should I just put in the form of a circle
Hi rock.freak667! :smile:

Yes, that's messy! :biggrin:

Definitely put it in the form of a nice circle (x² + (y-a)² = b²)! :wink:
 

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