https://file.aiquickdraw.com/mathbot/file/1726397756723rsbl36iz.jpg

The equation given is:

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

We are told that the equation represents a circle. Let's work through the problem step-by-step:

### Step 1: Simplify the equation

First, divide the entire equation by 8 to simplify:

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

### Step 2: Complete the square for both \(x\) and \(y\)

We need to complete the square to rewrite the equation in the standard form of a circle, \((x - h)^2 + (y - k)^2 = r^2\).

#### For \(x^2 + 14px\):

- Take half of the coefficient of \(x\), which is \(14p\), and square it: 
  \[
  \left(\frac{14p}{2}\right)^2 = (7p)^2 = 49p^2
  \]
- Add and subtract \(49p^2\) to complete the square:
  \[
  x^2 + 14px = (x + 7p)^2 - 49p^2
  \]

#### For \(y^2 - 8py\):

- Take half of the coefficient of \(y\), which is \(-8p\), and square it:
  \[
  \left(\frac{-8p}{2}\right)^2 = (-4p)^2 = 16p^2
  \]
- Add and subtract \(16p^2\) to complete the square:
  \[
  y^2 - 8py = (y - 4p)^2 - 16p^2
  \]

### Step 3: Rewrite the equation

Substitute the completed square expressions back into the equation:

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

Combine the constants:

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

Add \(65p^2\) to both sides:

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

### Step 4: Identify the radius

This is now in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\), where the radius is \(r = \sqrt{9p^2} = 3p\).

### Step 5: Answer the question

The length of the radius is given as \(np\), where \(n\) and \(p\) are positive constants. Comparing \(3p\) to \(np\), we can conclude that \(n = 3\).

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

---

Would you like any further details or have additional questions? Here are five related questions you might find useful:

1. How do you complete the square for any general quadratic equation?
2. What is the standard form of a circle and how do you derive it?
3. How can you find the center and radius of a circle from a general second-degree equation?
4. What is the geometric meaning of completing the square in relation to conic sections?
5. How would the equation change if the graph was an ellipse instead of a circle?

**Tip:** Completing the square is an essential technique when dealing with equations of conic sections like circles and ellipses. Mastery of this method is crucial for understanding more complex geometrical shapes.

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 by completing the square to find the value of n, where the radius is given as np. This step-by-step solution covers important concepts in circle geometry and quadratic completion for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation