The circle passes through the points (-4,8),(4,-4) and (3,-9) find it's equation

To find the equation of the circle passing through the points \((-4, 8)\), \((4, -4)\), and \((3, -9)\), we can use the general equation of a circle:

\[
x^2 + y^2 + Dx + Ey + F = 0
\]

where \(D\), \(E\), and \(F\) are constants that we need to determine.

### Step 1: Set up the system of equations
Substitute each of the three points into the equation to get three equations:

1. Substituting \((-4, 8)\):
   \[
   (-4)^2 + 8^2 + D(-4) + E(8) + F = 0
   \]
   Simplifying:
   \[
   16 + 64 - 4D + 8E + F = 0 \quad \Rightarrow \quad -4D + 8E + F = -80 \quad \text{(Equation 1)}
   \]

2. Substituting \((4, -4)\):
   \[
   (4)^2 + (-4)^2 + D(4) + E(-4) + F = 0
   \]
   Simplifying:
   \[
   16 + 16 + 4D - 4E + F = 0 \quad \Rightarrow \quad 4D - 4E + F = -32 \quad \text{(Equation 2)}
   \]

3. Substituting \((3, -9)\):
   \[
   (3)^2 + (-9)^2 + D(3) + E(-9) + F = 0
   \]
   Simplifying:
   \[
   9 + 81 + 3D - 9E + F = 0 \quad \Rightarrow \quad 3D - 9E + F = -90 \quad \text{(Equation 3)}
   \]

### Step 2: Solve the system of equations
We now have the following system of linear equations:

\[
\begin{aligned}
-4D + 8E + F &= -80 \quad \text{(1)}\\
4D - 4E + F &= -

Learn how to find the equation of a circle passing through the points (-4, 8), (4, -4), and (3, -9) using the general equation of a circle. This problem involves analytic geometry and circle equations, suitable for advanced high school students.

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