What loci is represented by the following equation

[thomas49th]

What loci is represented by the following equation:
|z+1| = |z-1|

I believe I can get the answer, I am just slightly confused by what is going on?

I turned this to

|z+1| / |z-1| = 1, which means |z+1| = 1 and |z-1| = 1. Doe this mean we can draw 2 circles of radius 1 at the points (1,0) and (-1,0)? The only place they touch is the origin? Is this correct?

OR
Letting z = x + iy

$$\frac{\sqrt{(x+1)^{2} + y^{2}}}{\sqrt{(x-1)^{2}+y^{2}}} = 1$$

$$\frac{(x+1)^{2} + y^{2}}{(x-1)^{2}+y^{2}} = 1$$

$$4x = 0$$

x = 0
That implies y can be anypoint along the y-axis (at x= 0). Is this correct?

Finally are the actual locations of the loci origins actually -1 and 1, because surely z doesn't have to be x + iy, I could be 5x + 6iy

Thanks
Thomas

 

[Quinzio]

That implies y can be anypoint along the y-axis (at x= 0). Is this correct?

yes.

Finally are the actual locations of the loci origins actually -1 and 1, because surely z doesn't have to be x + iy, I could be 5x + 6iy

?

 

[thomas49th]

yes.


?

Okay, forget the last bit, but why is my method of drawing circles and finding where they cross not valid?

 

[Quinzio]

Okay, forget the last bit, but why is my method of drawing circles and finding where they cross not valid?

You're given

a= b

You wrote
a/b = 1 which is correct.

Then you wrote
a=1, b=1
why ?
a=3, b=3 is valid, too, any combination of a = b is valid.
It's ok to cross circles, but their radious can be any number.

 

[thomas49th]

ahhhhhhhhhhhh I think I seeeeee
Cheers