the parabola has the equation y^(2)-6y+24x-111= 0 
find the distance between the focus and the origin power of 2

To find the distance between the focus of the parabola and the origin, we first need to rewrite the given equation in a more recognizable form.

The given equation is:

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

We will complete the square for the \( y \)-terms to rewrite this as the equation of a parabola.

### Step 1: Completing the Square for the \( y \)-terms

Starting with the terms involving \( y \):

\[
y^2 - 6y
\]

To complete the square, take half of the coefficient of \( y \) (which is \( -6 \)), square it, and add and subtract this value. Half of \( -6 \) is \( -3 \), and \( (-3)^2 = 9 \). So:

\[
y^2 - 6y = (y - 3)^2 - 9
\]

Now substitute this back into the original equation:

\[
(y - 3)^2 - 9 + 24x - 111 = 0
\]

Simplifying:

\[
(y - 3)^2 + 24x - 120 = 0
\]

Move the constant term to the other side:

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

Rearrange this into a standard form:

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

This is the equation of a horizontally oriented parabola, which has the standard form:

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

where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus (and to the directrix). Comparing the given equation with the standard form, we see:

- \( h = 5 \)
- \( k = 3 \)
- \( 4p = -24 \)

Thus, \( p = -6 \). The negative sign indicates that the parabola opens to the left.

### Step 2: Finding the Coordinates of the Focus

The vertex of the parabola is \( (5, 3) \). Since the parabola opens to the left and \( p = -6 \), the focus is located 6 units to the left of the vertex. So, the coordinates of the focus are:

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

### Step 3: Calculating the Distance Between the Focus and the Origin

The focus is at \( (-1, 3) \), and the origin is at \( (0, 0) \). The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

Substituting the coordinates of the focus and the origin:

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

### Step 4: Squaring the Distance

The square of the distance is:

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

Thus, the square of the distance between the focus and the origin is \( \boxed{10} \).

Learn how to find the distance between the focus of a parabola and the origin using the given equation y^2 - 6y + 24x - 111 = 0. This detailed step-by-step solution covers completing the square, determining the focus coordinates, applying the distance formula, and calculating the squared distance. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation