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.