Hey guys, could someone explain this answer (the correct answer is in red)? I understand that the region defined by |z|= 2 is a circle with r=2. By why is |z| >or= |z-(-sqrt(3) + i ) | defined by that line? And why is the required region above the line?
Thanks!
So we have
bit. If we had
That's just the perpendicular bisector of the line joining
)
and the origin.
With perpendicular bisectors, each point on it is the same distance from the end points of the line that it's cutting. I think I've linked this before, but I like the picture here
http://mathworld.wolfram.com/PerpendicularBisectorTheorem.html. That's what
is saying isn't it? We have the line from the origin to
- so those are our end points. So z being a set of complex numbers, |z| is the distance of some complex number from the origin, and it happens to be equal to the distance between some complex number and the point
. Hence that's why we know it's that line. (I hope that reasoning is okay, hopefully someone else will be able to confirm).
That's what the parts of the question before this one was pushing you towards. If you couldn't recognise that straight away, the question before had you state that
is equivalent to
(which you could have shown by subbing in z=x+yi, but that'd just be going the long way around).
Now note the
greater than, that's why we're looking at the region that's above that line.
What about the
bit? Well that's just a circle with radius 2.
Now the
--> intersection. The region we want is the region when
and
overlap.
Home
Help
Search
Login
