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The problem you're working on asks to find the coefficients \(a\), \(b\), \(c\), and \(d\) that satisfy the equation of a circle:

\[
ax^2 + ay^2 + bx + cy + d = 0
\]

From the image, we have a circle with several points given on its boundary:
- (-2, 7)
- (-4, 5)
- (-4, -3)

These points can be plugged into the general equation of the circle to form a system of linear equations, which we can solve using Gaussian-Jordan elimination.

### Steps to solve:

1. **Substitute the points into the circle's equation:**

   For \((-2, 7)\):
   \[
   a(-2)^2 + a(7)^2 + b(-2) + c(7) + d = 0
   \]
   Simplifies to:
   \[
   4a + 49a - 2b + 7c + d = 0
   \]
   \[
   53a - 2b + 7c + d = 0 \tag{1}
   \]

   For \((-4, 5)\):
   \[
   a(-4)^2 + a(5)^2 + b(-4) + c(5) + d = 0
   \]
   Simplifies to:
   \[
   16a + 25a - 4b + 5c + d = 0
   \]
   \[
   41a - 4b + 5c + d = 0 \tag{2}
   \]

   For \((-4, -3)\):
   \[
   a(-4)^2 + a(-3)^2 + b(-4) + c(-3) + d = 0
   \]
   Simplifies to:
   \[
   16a + 9a - 4b - 3c + d = 0
   \]
   \[
   25a - 4b - 3c + d = 0 \tag{3}
   \]

Now, we have the system of equations:
\[
53a - 2b + 7c + d = 0 \tag{1}
\]
\[
41a - 4b + 5c + d = 0 \tag{2}
\]
\[
25a - 4b - 3c + d = 0 \tag{3}
\]

2. **Solve the system using Gaussian-Jordan elimination** to find the values of \(a\), \(b\), \(c\), and \(d\).

I will now solve this system.The solution to the system of equations is as follows:

\[
a = \frac{-d}{47}, \quad b = \frac{4d}{47}, \quad c = \frac{2d}{47}
\]

The equation of the circle will therefore depend on the value of \(d\), and can be written as:

\[
\frac{-d}{47}x^2 + \frac{-d}{47}y^2 + \frac{4d}{47}x + \frac{2d}{47}y + d = 0
\]

This represents the general form of the circle equation. You can choose a specific value for \(d\) (typically \(d = 1\) or another constant) to fully determine the equation of the circle.

Would you like me to proceed with a specific value for \(d\) or do you have any other questions?

---

Here are 5 related questions to deepen your understanding:
1. How does changing the value of \(d\) affect the equation of the circle?
2. What is the significance of the coefficients \(a\), \(b\), and \(c\) in the circle equation?
3. How does the Gaussian-Jordan elimination method simplify solving systems of equations?
4. How would the circle equation change if another point on the circle were provided?
5. Can you generalize this method to find the equation of any conic section?

**Tip**: Gaussian-Jordan elimination is particularly useful when solving systems of linear equations because it provides a systematic approach to row-reducing matrices into their simplest forms.

Carilah koefisien a, b, c, dan d yang memenuhi persamaan lingkaran ax^2 + ay^2 + bx + cy + d = 0, dengan titik-titik (-2, 7), (-4, 5), dan (-4, -3) pada lingkaran.

Learn how to find the coefficients a, b, c, and d for the circle equation ax^2 + ay^2 + bx + cy + d = 0 using Gaussian-Jordan elimination. This example uses points (-2, 7), (-4, 5), and (-4, -3) that lie on the circle. Suitable for Grades 10-12.

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