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The given equation of the parabola is:

\[
y^2 - 6y + 24x - 111 = 0
\]

The task is to find the distance between the focus and the origin, squared.

### Step 1: Rewrite the Parabola Equation in Standard Form
We need to express the given equation in the standard form of a parabola.

Start by isolating the \(y\) terms on one side and complete the square for the \(y\)-terms:

\[
y^2 - 6y = -24x + 111
\]

To complete the square:
1. Take half of the coefficient of \(y\) (which is \(-6\)), divide by 2, and square it: 
   \(\left(\frac{-6}{2}\right)^2 = 9\).

2. Add and subtract 9 on the left-hand side:

\[
y^2 - 6y + 9 - 9 = -24x + 111
\]

This simplifies to:

\[
(y - 3)^2 - 9 = -24x + 111
\]

Add 9 to both sides to balance the equation:

\[
(y - 3)^2 = -24x + 120
\]

Now, express it in the standard form of a horizontal parabola:

\[
(y - 3)^2 = -24(x - 5)
\]

This represents a parabola with vertex \((5, 3)\) that opens to the left.

### Step 2: Identify the Key Parameters
For a parabola in the form \((y - k)^2 = 4p(x - h)\), the distance from the vertex to the focus is \(|p|\).

From the equation \((y - 3)^2 = -24(x - 5)\), we can see that \(4p = -24\), so:

\[
p = \frac{-24}{4} = -6
\]

The focus is located 6 units to the left of the vertex at \((5 - 6, 3) = (-1, 3)\).

### Step 3: Calculate the Distance from the Focus to the Origin
The focus is at \((-1, 3)\), and the origin is at \((0, 0)\). The distance \(d\) between the focus and the origin is calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates \((-1, 3)\) and \((0, 0)\):

\[
d = \sqrt{(-1 - 0)^2 + (3 - 0)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

### Step 4: Square the Distance
The problem asks for the distance squared, so:

\[
d^2 = (\sqrt{10})^2 = 10
\]

### Final Answer:
The square of the distance between the focus and the origin is \( \boxed{10} \).

Would you like a detailed explanation of any step, or have more questions?

### Further Questions:
1. How do you derive the equation of a parabola from its geometric definition?
2. What are the other forms of the parabola equation, and when are they used?
3. Can you explain how completing the square helps to rewrite quadratic equations?
4. How does the distance formula apply to various geometric problems?
5. How do we locate the directrix of a parabola given its equation?

### Tip:
When dealing with conic sections, always try to rewrite the equation into a recognizable standard form, as it helps in identifying key features like the vertex, focus, and directrix.

The parabola has the equation y^2 - 6y + 24x - 111 = 0. Find the distance between the focus and the origin, squared.

Solve the problem of finding the distance between the focus and the origin for the given parabola equation y^2 - 6y + 24x - 111 = 0. Learn the steps to rewrite the equation in standard form, calculate the focus, and use the distance formula. Suitable for students in Grades 10-12.

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