A straight line in the complex plane

[rajeshmarndi]

Homework Statement

w+w* = (s+t*)z + (t+s*)z* + r+r* = 0, is a straight line. Then couldn't find how does
w-w* = (s-t*)z + (t-s*)z* + r-r* = 0, is also a straight line.

Relevant Equations

w = sz+tz*+r=0

sz+tz*+r=0=say w

so w* = s*z* + t*z + r*=0

Now ,
w+w* = (s+t*)z + (t+s*)z* + r+r* = 0
= p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w)
which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number.

Now, again,
w-w* = (s-t*)z + (t-s*)z* + r-r* = 0

I couldn't understand, in the solution, how this is also termed as a complex straight line like eq(1).
Since when this is worked out, it comes to be as,

q*z - qz* + id = 0 ( since r-r* will give imaginary number)

This is not in the form of a complex straight line equation.

Thanks.

 

[Ray Vickson]

sz+tz*+r=0=say w

so w* = s*z* + t*z + r*=0

Now ,
w+w* = (s+t*)z + (t+s*)z* + r+r* = 0
= p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w)
which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number.

Now, again,
w-w* = (s-t*)z + (t-s*)z* + r-r* = 0

I couldn't understand, in the solution, how this is also termed as a complex straight line like eq(1).
Since when this is worked out, it comes to be as,

q*z - qz* + id = 0 ( since r-r* will give imaginary number)

This is not in the form of a complex straight line equation.

Thanks.

What, in plain language, is the statement of the problem? Is it "prove that w+w*=0 describes a straight line in the complex plane"? Is it something else? I cannot figure out what you want.

 

[rajeshmarndi]

What, in plain language, is the statement of the problem?

I want to know,
given w= sz+tz*+r=0

Is
w-w* = (s-t*)z + (t-s*)z* + r-r* = 0
also a complex straight line?

[edit: r,s,t are non-zero complex number and z=x+iy (x,y ε R) ]

 

[Ray Vickson]

I want to know,
given w= sz+tz*+r=0

Is
w-w* = (s-t*)z + (t-s*)z* + r-r* = 0
also a complex straight line?

[edit: r,s,t are non-zero complex number and z=x+iy (x,y ε R) ]

Write $w$ in real terms, and expand it out to see what you get. That is, write $s=s_1+i s_2, t = t_1+i t_2, r = r_1+i r_2$ and $z = x + i y$.

 

[rajeshmarndi]

Write $w$ in real terms, and expand it out to see what you get. That is, write $s=s_1+i s_2, t = t_1+i t_2, r = r_1+i r_2$ and $z = x + i y$.

$w$ becomes,

$w = [(s_1+t_1)x + (t_2-s_2)y] + i[(s_2+t_2)x + (s_1-t_1)y+r_2]=0$

So the real terms of $w$ is $[(s_1+t_1)x + (t_2-s_2)y]=0$