Write the given equation of a straight line in complex notation:
Straight line through 1 and (-1 – i)
Attempt:
So i treated this initially just like a set of coordinates in the set of R thus (1,0) and (-1,-1) and solved for an equation of the line:
Y = x/2 – 1/2
Now keeping in mind that i know to express a line in complex form it is going to have to look like some form of Re(az+b) = 0, where a,b are complex values
I manipulated it to: x – 2y – 1 = 0
Now here is where i am stuck. I know x-2y is the real part of some complex number, but i can't figure out what or how to obtain it. All i know is that the product of the complex numbers a and z has to equal x-2y in the real portion of it at least.
Suggestions?
Best Answer
In the equation $x-2y-1=0$ replace $x,y$ by their expressions $x=\frac {z+\bar z}{2}, y=\frac {z-\bar z}{2i}$in terms of $z,\bar z$ and obtain after an easy calculation $(1+2i)z+(1-2i)\bar z-2=0$ .