Circle

Let the equation of circle be:
x2 + y2 + Dx + Ey + F = 0
Sub ( 0 , - 1 ),
02 + ( - 1 )2 + D ( 0 ) + E ( - 1 ) + F = 0
F = E - 1 --- ( 1 )
Sub x = 1,
12 + y2 + D + Ey + F = 0
y2 + Ey + ( D + F + 1 ) = 0
As the circle touches the line, delta = 0, i.e.
E2 - 4 ( 1 )( D + F + 1 ) = 0
E2 = 4D + 4F + 4 --- ( 2 )
Similarly, sub y = 1,
x2 + Dx + ( E + F + 1 ) = 0
Delta = 0, i.e.
D2 - 4 ( 1 )( E + F + 1 ) = 0
D2 = 4E + 4F + 4 --- ( 3 )
Put ( 1 ) into ( 2 ),
E2 = 4D + 4 ( E - 1 ) + 4
E2 = 4D + 4E --- ( 4 )
Put ( 1 ) into ( 3 ),
D2 = 4E + 4 ( E - 1 ) + 4
E = D2 / 8 --- ( 5 )
Put ( 5 ) into ( 4 ),
( D2 / 8 )2 = 4D + 4 ( D2 / 8 )
D ( D3 - 32D - 256 ) = 0
D ( D - 8 )( D2 + 8D + 32 ) = 0
D = 0 or 8
E = 0 or 8
F = -1 or 7
Therefore the equations:
x2 + y2 - 1 = 0 or x2 + y2 + 8x + 8y + 7 = 0

The unit circle fits this all.
Three constraints to three variables make the solution unique.