1. ## Complex and Geometry

What does equation indicate on the plane?

2. Hello, dhiab!

What does this equation indicate on the plane: .$\displaystyle z + \overline{z} \:=\:k|z|$

. . where: .$\displaystyle z \:=\:x + iy,\quad \overline{z} \:=\:x - iy, \quad k \in \mathbb{R}$
The left side is: .$\displaystyle z + \overline{z} \:=\:(x+iy) + (x - iy) \:=\:2x$

The right side is: .$\displaystyle k|z| \:=\:k\sqrt{x^2+y^2}$

We have: .$\displaystyle 2x \:=\:k\sqrt{x^2+y^2}$

Square both sides: .$\displaystyle 4x^2 \:=\:k^2x^2 + k^2y^2 \quad\Rightarrow\quad 4x^2 - k^2x^2 \:=\:k^2y^2$

. . $\displaystyle (4-k^2)x^2 \:=\:k^2y^2 \quad\Rightarrow\quad \frac{4-k^2}{k^2}x^2 \:=\:y^2$

And we have: .$\displaystyle y \;=\;\pm\frac{\sqrt{4-k^2}}{k}\,x$

It is a pair of intersecting lines.

. . Note that: .$\displaystyle |k| \:\leq\: 2\:\text{ and }\:k \:\neq\: 0$

3. Originally Posted by Soroban
Hello, dhiab!

The left side is: .$\displaystyle z + \overline{z} \:=\x+iy) + (x - iy) \:=\:2x$

The right side is: .$\displaystyle k|z| \:=\:k\sqrt{x^2+y^2}$

We have: .$\displaystyle 2x \:=\:k\sqrt{x^2+y^2}$

Square both sides: .$\displaystyle 4x^2 \:=\:k^2x^2 + k^2y^2 \quad\Rightarrow\quad 4x^2 - k^2x^2 \:=\:k^2y^2$

. . $\displaystyle (4-k^2)x^2 \:=\:k^2y^2 \quad\Rightarrow\quad \frac{4-k^2}{k^2}x^2 \:=\:y^2$

And we have: .$\displaystyle y \;=\;\pm\frac{\sqrt{4-k^2}}{k}\,x$

It is a pair of intersecting lines.

. . Note that: .$\displaystyle |k| \:\leq\: 2\:\text{ and }\:k \:\neq\: 0$

HELLO : Thank you
I'think x, k for same sign , you have the segment is not lines .