Property 12
\(z=x+iy,\ z_1=a+ib\)
\(|z-z_1|=r\iff(x-a)^2+(y-b)^2=r^2\)
The locus is a circle with center \((a,b)\) and radius \(r\).
1.
\(|z-1+i|=3\)
2.
\(|z-1-2i|=1\)
3.
\(|z-2-2i|<1\)
4.
\(|z+3|\ge2\)
5.
\(1<|z+i|\le2\)
Property 13
\(z_1=a+ib,\ z_2=c+id\)
Distance: \(|z_1-z_2|\).
Equal-distance loci form a line.
For two circles, shortest distance is \(|O_1O_2|-r_1-r_2\); longest distance is \(|O_1O_2|+r_1+r_2\).
1.
\(z=x+iy\)
\(|z-1|=|z+i|\)
\(\Rightarrow ?\)
2.
\(|z|=|z-2i|\)
3.
\(|z-2|=|z+i|\)
4.
\(|z-1+i|=|z+2i|\)
5.
\(|z-1|\le|z|\)
6.
\(|z+1|>|z-i+1|\)
7.
\(|z-2|>|z-3i|\)
8.
\(|z+1|<|z-i|\)
9.
\(|z+i|=|z-1|\)
10.
\(|z-i+1|=|z-2|\)
11.
\(|z-2i|\le|z+2|\)
12.
\(|z|>|z+i|\)
13.
\(|z-2|<|z-1+i|\)
14.
\(|z|\le3\)
\(|z-(5+12i)|=r\)
\(\Rightarrow\max(r)=?\)
15.
\(|z|\le3\)
\(|z-(5+12i)|=r\)
\(\Rightarrow\min(r)=?\)
16.
\(|z_1-(4+3i)|=2\)
\(|z_2-(1-i)|=1\)
\(\Rightarrow\min|z_1-z_2|=?\)
17.
\(|z_1-(4+3i)|=2\)
\(|z_2-(1-i)|=1\)
\(\Rightarrow\max|z_1-z_2|=?\)