Step 1: General circle
Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. Passing through $(0,1)$: $1 + 2f + c = 0$. Passing through $(1,1)$: $2 + 2g + 2f + c = 0$.
Step 2: Tangency
The tangent at $(1,1)$ is $y - 1 = 2(x - 1)$, i.e. $2x - y - 1 = 0$. The tangent to the circle at $(1,1)$ is $x + y + g(x+1) + f(y+1) + c = 0$, with coefficient ratio matching: $(1 + g) : (1 + f) = 2 : -1$, so $1 + f = -\frac{1}{2}(1 + g)$.
Step 3: Solve
Subtracting the two point equations gives $1 + 2g = 0$, so $g = -\frac{1}{2}$. Then $1 + f = -\frac{1}{2}\cdot\frac{1}{2}$, so $f = -\frac{5}{4}$.
Step 4: Centre
The centre is $(-g, -f) = \left(\frac{1}{2}, \frac{5}{4}\right)$, which matches option (C).
Final Answer:
The centre is (1/2, 5/4). This is option (C).
\[ \boxed{\text{(C) }\left(\frac{1}{2},\frac{5}{4}\right)} \]