Step 1: Shift the origin to the meeting point:
Put $X = x - 1$, $Y = y$. The lines become $X + Y = 0$ and $X - Y = 0$, and the original origin is at $(-1, 0)$ in the new frame.
Step 2: Circles touching both lines:
The two lines cross at right angles and their bisectors are the new axes, so the centre of a touching circle lies on an axis. A centre $(c, 0)$ or $(0, c)$ is at distance $|c|/\sqrt{2}$ from both lines, so that is the radius.
Step 3: Impose the point $(-1,0)$:
Centre $(c,0)$: $(c+1)^2 = c^2/2$, so $c^2 + 4c + 2 = 0$, with discriminant $16 - 8 = 8 > 0$, giving 2 values.
Centre $(0,c)$: $1 + c^2 = c^2/2$, so $c^2 = -2$, impossible.
Step 4: Count:
Exactly 2 circles.
Final Answer:
Option (B), 2 circles.
\[ \boxed{2 \text{ (B)}} \]