To find the value of [tex]\( n \)[/tex] such that the radius of the circle is [tex]\( np \)[/tex], we need to rewrite the given equation in the standard form of a circle, which is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
Given equation:
[tex]\[
8x^2 + 112px + 8y^2 - 64py = -448p^2
\][/tex]
First, divide the entire equation by 8 to simplify:
[tex]\[
x^2 + 14px + y^2 - 8py = -56p^2
\][/tex]
Next, we need to complete the square for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms:
Complete the square for the [tex]\( x \)[/tex] term:
[tex]\[
x^2 + 14px
\][/tex]
Add and subtract [tex]\((7p)^2\)[/tex] inside the equation:
[tex]\[
x^2 + 14px + (7p)^2 - (7p)^2 = (x + 7p)^2 - 49p^2
\][/tex]
Complete the square for the [tex]\( y \)[/tex] term:
[tex]\[
y^2 - 8py
\][/tex]
Add and subtract [tex]\((4p)^2\)[/tex] inside the equation:
[tex]\[
y^2 - 8py + (4p)^2 - (4p)^2 = (y - 4p)^2 - 16p^2
\][/tex]
Now, substitute these completed squares back into the equation:
[tex]\[
(x + 7p)^2 - 49p^2 + (y - 4p)^2 - 16p^2 = -56p^2
\][/tex]
Combine the constant terms on the left side:
[tex]\[
(x + 7p)^2 + (y - 4p)^2 - 65p^2 = -56p^2
\][/tex]
Move the constant term [tex]\( -65p^2 \)[/tex] to the right side:
[tex]\[
(x + 7p)^2 + (y - 4p)^2 = 65p^2 - 56p^2
\][/tex]
[tex]\[
(x + 7p)^2 + (y - 4p)^2 = 9p^2
\][/tex]
The equation [tex]\((x + 7p)^2 + (y - 4p)^2 = 9p^2\)[/tex] is in the standard form of a circle [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex] where the radius [tex]\( r \)[/tex] is [tex]\( \sqrt{9p^2} = 3p \)[/tex].
Therefore, the value of [tex]\( n \)[/tex] is:
[tex]\[
\boxed{3}
\][/tex]
