https://file.aiquickdraw.com/mathbot/file/1726680084421kimrtyok.jpg

The equation of the parabola given in the image is:

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

We can rewrite this into a standard form for a horizontal parabola, which will help us find the focus and other necessary components.

### Step 1: Complete the square
Start by grouping the terms involving \( y \):

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

Now complete the square for \( y^2 - 6y \). To complete the square, take half of the coefficient of \( y \), square it, and add it to both sides. Half of \( -6 \) is \( -3 \), and \( (-3)^2 = 9 \).

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

This simplifies to:

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

Factor the right-hand side:

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

### Step 2: Identify the vertex form
The equation is now in the form \( (y - k)^2 = 4p(x - h) \), which represents a horizontal parabola. From this equation, we identify:

- Vertex: \( (5, 3) \)
- \( 4p = -24 \), so \( p = -6 \)

### Step 3: Focus and Distance to the Origin
Since \( p = -6 \), the focus lies \( 6 \) units to the left of the vertex (since it's negative), at:

\[
(5 - 6, 3) = (-1, 3)
\]

The origin is at \( (0, 0) \), and the distance between the focus \( (-1, 3) \) and the origin is:

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

### Step 4: Distance Squared
The problem asks for the square of this distance:

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

Thus, the squared distance between the focus and the origin is \( 10 \).

Would you like further details on any of these steps?

---

Here are 5 related questions to extend your understanding:
1. How do you derive the vertex form of a parabola from the general quadratic equation?
2. How is the distance formula used to calculate the distance between two points?
3. What is the significance of the value of \( p \) in a parabola?
4. How do you know when a parabola opens horizontally or vertically?
5. How would you find the directrix of this parabola?

**Tip:** Completing the square is a useful technique not only for conic sections but also for solving quadratic equations and optimizing functions.

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 involves key concepts in conic sections, completing the square, and applying the distance formula. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation