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We are given the equation:

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

We need to determine the value of \( n \) when the graph of this equation represents a circle, and the radius of the circle is \( np \), where \( n \) and \( p \) are positive constants.

### Step 1: Simplify the equation

First, divide the entire equation by 8 to make the coefficients of \( x^2 \) and \( y^2 \) equal to 1:

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

### Step 2: Completing the square

Now, we complete the square for both the \( x \) and \( y \) terms.

#### For the \( x \)-terms:
We have \( x^2 + 14px \). To complete the square, we take half of 14p, which is \( 7p \), and square it:

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

So, \( x^2 + 14px \) becomes \( (x + 7p)^2 - 49p^2 \).

#### For the \( y \)-terms:
We have \( y^2 - 8py \). To complete the square, we take half of -8p, which is \( -4p \), and square it:

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

So, \( y^2 - 8py \) becomes \( (y - 4p)^2 - 16p^2 \).

### Step 3: Substitute back into the equation

Substituting the completed squares into the equation:

\[
(x + 7p)^2 - 49p^2 + (y - 4p)^2 - 16p^2 = -56p^2
\]

Combine like terms:

\[
(x + 7p)^2 + (y - 4p)^2 - 65p^2 = -56p^2
\]

Move the constants to the right-hand side:

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

### Step 4: Identify the radius

This is the equation of a circle with center \( (-7p, 4p) \) and radius \( 3p \) (since the right-hand side is \( 9p^2 \), the radius is \( \sqrt{9p^2} = 3p \)).

### Step 5: Determine \( n \)

The length of the radius is given by \( np \). Since we have found that the radius is \( 3p \), we can conclude that \( n = 3 \).

Thus, the value of \( n \) is:

\[
\boxed{3}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. What is the general equation of a circle in the coordinate plane?
2. How do you complete the square for quadratic terms involving a variable?
3. How does the process of completing the square help in identifying the center of a circle?
4. How do you determine the radius from the equation of a circle?
5. What are the conditions that define an equation as representing a circle?

### Tip:
When solving for the equation of a circle, always look to complete the square to transform the quadratic terms into a recognizable circle equation format.

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?

Learn how to solve for the radius of a circle given its equation in standard form. This solution demonstrates the process of completing the square and extracting key circle properties, suitable for high school students (Grades 9-12).

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