4. Find the geometric locus of points on the coordinate plane, the coordinates \(x\) and \(y\) of which satisfy one of the following conditions:
1) \(y^{2}-x^{2}=0\)
3) \(x^{2}+y^{2}=4(y-1)\)
2) \(x^{2}-2 x+1=0\)
4) \(x^{2}-2 x y+y^{2}=-1\).
4. 1) \(\mathrm{P}\) is a solution. Since \((y-x)(y+x)=0\), then \(y=x\) or \(y=-x\). Therefore, the desired set of points will be the union of the sets of points of the graphs of these functions.

2) Points of the line \(x=1.3\) ) Point \(A(0 ; 2)\).

4) Empty set.