complex numbers (1 Viewer)

[maths lover]

i am having troubles sketching the locus of this inequality to find the range of modulus. the question is modulus of(z-sqrt(3)-i)=2. i might be brain dead or something but why does this graph cut at the origin, thanks.

 

[SpiralFlex]

Let us interpret this result geometrically then algebraically,



So geometrically, we know this is a circle with a radius of 2. What this means is how far is the locus of z from the point (Centre)

So let us draw our pretty diagram,



And there you have it.



Algebraically we let,

 

[maths lover]

yeah i did all that i just dont get why it crosses the origin. sorry.
EDIT: i understand that you can use Pythagoras to see that it cuts the origin, but how at first glance did you know it cut the origin.

 

[SpiralFlex]

For the locus of z to cross the origin.



Has to be true for point





Ah, it's true. Therefore it crosses the origin.

If you have a curve that is close to the origin and you do not know if it crosses it, there is no harm in testing. Only takes 1 micro-second.

 

[maths lover]

oh ok i was brain dead, since i didnt find the equation of the circle just geometrically interpreted and ended up drawing it so that it didnt cut the origin. thanks spiral for everything.