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To solve the problem, we first need to rewrite the equation of the parabola in its standard form.

### Given Equation
The equation of the parabola is:

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

### Step 1: Convert to Standard Form
This is a parabola that opens to the right or left because it has \(y^2\) as the squared term. Let's rewrite the equation in the form:

\[
(y - k)^2 = 4p(x - h)
\]

1. **Isolate the \(y\)-terms:**

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

2. **Complete the square for the \(y\)-terms:**

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

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

Now the equation is in the standard form \((y - k)^2 = 4p(x - h)\), where:
- \(h = 5\)
- \(k = 3\)
- \(4p = -24 \Rightarrow p = -6\)

### Step 2: Find the Focus
The vertex of the parabola is at \((h, k) = (5, 3)\). Since \(p = -6\), the parabola opens to the left (because \(p < 0\)).

The focus of the parabola will be at \((h + p, k)\):
\[
\text{Focus} = (5 - 6, 3) = (-1, 3)
\]

### Step 3: Find the Distance Between the Focus and the Origin
Now, we need to find the distance between the focus \((-1, 3)\) and the origin \((0, 0)\). The distance formula is:

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

Substitute the coordinates:

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

### Step 4: Find the Distance Squared
Since the question asks for the distance "power of 2," we need to find the square of this distance:

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

### Final Answer
The distance squared between the focus and the origin is **10**.

---

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How do you complete the square for quadratic expressions?
2. What are the different standard forms of a parabola, and how are they derived?
3. How do you find the vertex, focus, and directrix of a parabola?
4. How does the value of \(p\) affect the shape and orientation of a parabola?
5. Can you determine the equation of a parabola given its vertex and focus?

**Tip:** Remember that completing the square is a common technique to rewrite quadratic equations in standard form, which makes it easier to analyze their properties.

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

Learn how to find the distance between the focus and the origin for the parabola y^2 - 6y + 24x - 111 = 0. This problem explores the standard form of parabolas, completing the square, and applying the distance formula. Suitable for Grades 10-12.

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