https://file.aiquickdraw.com/mathbot/file/1727782187243enadzqxw.jpg

The given equation is:

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

The equation of a circle in standard form is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center and \(r\) is the radius.

### Step 1: Divide by 8
First, let's divide the entire equation by 8 to simplify it:

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

### Step 2: Complete the square
Now, we'll complete the square for both \(x\) and \(y\).

#### For \(x\):
The \(x\)-terms are \(x^2 + 14px\).
To complete the square, take half of the coefficient of \(x\), square it, and add it:
\[
\left(\frac{14p}{2}\right)^2 = (7p)^2 = 49p^2
\]
So, we add and subtract \(49p^2\):

\[
x^2 + 14px + 49p^2
\]

#### For \(y\):
The \(y\)-terms are \(y^2 - 8py\).
Similarly, complete the square for \(y\) by taking half of \(-8p\), squaring it, and adding it:
\[
\left(\frac{-8p}{2}\right)^2 = (-4p)^2 = 16p^2
\]
So, we add and subtract \(16p^2\):

\[
y^2 - 8py + 16p^2
\]

### Step 3: Rewriting the equation
Now, the equation becomes:

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

Simplifying this:

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

Move the constants to the right side:

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

### Step 4: Identify the radius
This is now in the form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\), where the radius \(r\) is:

\[
r = \sqrt{9p^2} = 3p
\]

### Step 5: Solving for \(n\)
We are given that the radius is \(np\), so:

\[
np = 3p
\]

Thus, \(n = 3\).

### Final Answer:
The value of \(n\) is \( \boxed{3} \).

---

Would you like a more detailed explanation of any part, or do you have any questions?

Here are some related questions for further exploration:
1. How do you complete the square for general quadratic equations?
2. What is the geometric interpretation of completing the square in the context of circles?
3. Can the equation of a circle include coefficients other than 1 for \(x^2\) and \(y^2\)?
4. How would the equation change if the circle was centered at the origin?
5. What is the significance of the constant on the right-hand side of a circle's equation?

**Tip:** When completing the square, always remember to balance the equation by adding or subtracting the same constant from both sides.

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?

Discover how to find the value of n in a circle equation by completing the square. This problem demonstrates key algebraic techniques such as simplifying equations, completing the square, and solving for constants. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation