Step 1: Let the equation of the circle be

$$(x - h)^2 + (y - k)^2 = r^2$$

where centre is $(h, k)$ and radius is $r$.

Step 2: Since the circle passes through $(2, 3)$ and $(-1, 1)$, we have:

$$(2 - h)^2 + (3 - k)^2 = r^2 \quad \text{... (1)}$$ $$(-1 - h)^2 + (1 - k)^2 = r^2 \quad \text{... (2)}$$

Step 3: Equating (1) and (2) since both equal $r^2$:

$$(2 - h)^2 + (3 - k)^2 = (-1 - h)^2 + (1 - k)^2$$

Expanding:

$$4 - 4h + h^2 + 9 - 6k + k^2 = 1 + 2h + h^2 + 1 - 2k + k^2$$

Simplifying:

$$6h + 4k = 11 \quad \text{... (3)}$$

Step 4: Since centre lies on line $x - 3y - 11 = 0$:

$$h - 3k = 11 \quad \text{... (4)}$$

Step 5: Solving equations (3) and (4):

From (4):

$$h = 11 + 3k$$

Substituting in (3):

$$6(11 + 3k) + 4k = 11$$ $$66 + 18k + 4k = 11$$ $$22k = -55, \quad \text{so} \quad k = -\frac{5}{2}$$

Then

$$h = 11 + 3\left(-\frac{5}{2}\right) = \frac{7}{2}$$

Step 6: Finding $r^2$ using point $(2, 3)$:

$$r^2 = \left(2 - \frac{7}{2}\right)^2 + \left(3 + \frac{5}{2}\right)^2 = \frac{9}{4} + \frac{121}{4} = \frac{65}{2}$$

Step 7: The equation of the circle is:

$$\left(x - \frac{7}{2}\right)^2 + \left(y + \frac{5}{2}\right)^2 = \frac{65}{2}$$