Complex Numbers: Equation involving the argument operator.

[Bradyns]

Homework Statement

Question:

Homework Equations

Any relevant to complex numbers.

The Attempt at a Solution

Given,

Arg(\frac{z}{w})= Arg(z)-Arg(w)

z=x+yi
z₁ = -1-2i
z₂ = 2+3i

Arg(z-z₁)=Arg(z₂-z₁)

LHS:
Arg(x+yi+1+2i)
Arg((x+1) + i(y+2))

tan(\theta)=\frac{y+2}{x+1}

RHS
Arg(2+3i+1+2i)
Arg(3+5i)

tan(\theta)=\frac{5}{3}

RHS = LHS

\frac{y+2}{x+1} = \frac{5}{3}

Simplified
y = \frac{5}{3}x -\frac{1}{3}

Is this right?

 

[Sunil Simha]

Yes it is correct.

 

[Bradyns]

Yes it is correct.

Cool beans,

Is there a simple way to solve these though?

A shortcut method?Wait, I found it...

It's the line created from the points z₁= -1-2i and z₂ = 2+3i

Gradient = \frac{3 - (-2)}{2 - (-1)} = \frac{5}{3}

y = \frac{5}{3}x + b

(-2) = \frac{5}{3}(-1) + b

-6 = -5 + 3b
-1 = 3b
b = -\frac{1}{3}

 

[Sunil Simha]

Yes. That's how I solved it. Your original method is still fine though. If you have loads of practice,, you'll be able to use both the methods equally efficiently.