Complex Help (1 Viewer)

[Lukybear]

If , where k is real

Find with aid of a diagram the value of |z|

 

[cyl123]

Change (z-1+i)/(z+1-i)=ki to (z-(1-i))/(z-(1+i))=ki --> (z-(1-i))=ki*(z-(1+i)) --> rotation by 90 degrees since k is real.

This means the angle between vectors (z-(1-i)) and (z-(-1+i)) is 90 degrees since k is real

This is same as sketching Arg[(z-1+i)/(z+1-i)]=pi/2 or -pi where the points 1-i and -1+i are included
Plotting the locus of z on Argand diagram we get the diagram below

In other words, z is a point on the circle centred at the origin, with the diameter being the line joining 1-i and -1+i.

k will only change the position of z on the circle but |z| should be sqrt(2)

 

[Affinity]

or you can:
z(1-ki) = (k-1) + (k+1)i
|z||1-ki| = |(k-1) + (k+1)i|
since k is real
|z|(sqrt(1+k^2)) = sqrt((k-1)^2+ (k+1)^2) = sqrt(2(k^2 + 1)) = sqrt(2) sqrt(k^2+1)
so |z| = sqrt(2)