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We are given the equation of a circle in a general quadratic form, and we need to find the value of \( n \) in the context of the problem. The equation provided is:

\[
8x^2 + 112px + 8y^2 - 64py = -448p^2
\]

Let's rewrite this equation in the standard form of a circle:

### Step 1: Group the terms involving \( x \) and \( y \)
\[
8x^2 + 112px + 8y^2 - 64py = -448p^2
\]

Divide the entire equation by 8 to simplify:

\[
x^2 + 14px + y^2 - 8py = -56p^2
\]

### Step 2: Complete the square
We will complete the square for both \( x \) and \( y \) terms.

#### Completing the square for \( x \)-terms:
We have \( x^2 + 14px \). To complete the square, take half of 14p (which is 7p), square it (giving \( 49p^2 \)), and add it to both sides.

\[
x^2 + 14px + 49p^2 = (x + 7p)^2
\]

#### Completing the square for \( y \)-terms:
We have \( y^2 - 8py \). To complete the square, take half of -8p (which is -4p), square it (giving \( 16p^2 \)), and add it to both sides.

\[
y^2 - 8py + 16p^2 = (y - 4p)^2
\]

Now, adding \( 49p^2 \) and \( 16p^2 \) to both sides of the equation:

\[
x^2 + 14px + y^2 - 8py + 49p^2 + 16p^2 = -56p^2 + 49p^2 + 16p^2
\]

\[
(x + 7p)^2 + (y - 4p)^2 = 9p^2
\]

### Step 3: Identify the radius
The standard form of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius. In our case, we have:

\[
(x + 7p)^2 + (y - 4p)^2 = (3p)^2
\]

So, the radius \( r = 3p \).

### Step 4: Find the value of \( n \)
We are told that the radius is \( np \), where \( n \) is a positive constant. Since we found that the radius is \( 3p \), it follows that:

\[
n = 3
\]

Thus, the value of \( n \) is \( \boxed{3} \).

Would you like further clarification on any part of the process? Here are five questions that could help expand your understanding:

1. How do you complete the square for different quadratic expressions?
2. Why do we divide the equation by 8 in the first step?
3. What is the general form of the equation of a circle?
4. How do you identify the center and radius from the standard form of a circle?
5. Can you derive a circle's equation from general quadratic forms involving \( x \) and \( y \)?

**Tip:** When dealing with equations of a circle, always aim to rearrange the equation into its standard form to easily identify the radius and center.

In the xy-plane, the graph of the given equation is a circle. The length of the radius of the circle is np, where n and p are positive constants. What is the value of n?

Solve a circle equation to find the value of n when the radius of a circle is given as np. Learn how to complete the square and convert the equation into standard form. Suitable for high school algebra students.

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