Given information:
The equation of the circle passing through (1,5) and (4,1) and touching the y-axis is x^2 + y^2 - 5x - 6y + 9k(4x + 3y - 19) = 0.

Approach:
To find the value of k, we need to use the given information about the circle and solve for k. Here's how we can approach the problem:

Step 1: Find the equation of the circle:
We are given that the circle passes through (1,5) and (4,1) and touches the y-axis. Since the circle touches the y-axis, its center lies on the x-axis. Let's assume the center of the circle is (h,0).

Using the distance formula, we can find the radius of the circle:
r = √[(4-1)^2 + (1-5)^2] = √[9 + 16] = √25 = 5

The equation of a circle with center (h,0) and radius r is given by:
(x-h)^2 + (y-0)^2 = r^2
(x-h)^2 + y^2 = 25

Step 2: Substitute the given points:
We are given that the circle passes through (1,5) and (4,1). Substituting these points into the equation of the circle, we get two equations:

(1-h)^2 + 5^2 = 25
(4-h)^2 + 1^2 = 25

Simplifying these equations, we get:
1 - 2h + h^2 + 25 = 25
16 - 8h + h^2 + 1 = 25

Simplifying further, we obtain:
h^2 - 2h + 1 = 0
h^2 - 8h + 16 = 0

Step 3: Solve for h:
We can solve the above two equations to find the value of h. Factoring the equations, we get:
(h - 1)^2 = 0
(h - 4)^2 = 0

From these equations, we can determine that h = 1 and h = 4.

Step 4: Substitute h into the equation of the circle:
Now that we have the values of h, we can substitute them into the equation of the circle to get two possible equations:

(x-1)^2 + y^2 = 25
(x-4)^2 + y^2 = 25

Step 5: Find the value of k:
Finally, we substitute the equations of the circle into the given equation and solve for k:

(x-1)^2 + y^2 - 5(x-1) - 6y + 9k(4(x-1) + 3y - 19) = 0
(x-4)^2 + y^2 - 5(x-4) - 6y + 9k(4(x-4) + 3y - 19) = 0

Simplifying these equations, we get two equations with k:

-9k + 4x -