Find the equation of the circle passing through (-3,6) , (-5,2) and (3, -6).

To find the equation of the circle passing through the points \((-3, 6)\), \((-5, 2)\), and \((3, -6)\), 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 need to be determined.

### Step 1: Substituting the points into the equation

We substitute each point into the general equation:

1. For \((-3, 6)\):
   \[
   (-3)^2 + (6)^2 + D(-3) + E(6) + F = 0
   \]
   \[
   9 + 36 - 3D + 6E + F = 0
   \]
   \[
   45 - 3D + 6E + F = 0
   \quad \text{(Equation 1)}
   \]

2. For \((-5, 2)\):
   \[
   (-5)^2 + (2)^2 + D(-5) + E(2) + F = 0
   \]
   \[
   25 + 4 - 5D + 2E + F = 0
   \]
   \[
   29 - 5D + 2E + F = 0
   \quad \text{(Equation 2)}
   \]

3. For \((3, -6)\):
   \[
   (3)^2 + (-6)^2 + D(3) + E(-6) + F = 0
   \]
   \[
   9 + 36 + 3D - 6E + F = 0
   \]
   \[
   45 + 3D - 6E + F = 0
   \quad \text{(Equation 3)}
   \]

### Step 2: Solving the system of equations

We now solve the system of equations formed by:

1. \( 45 - 3D + 6E + F = 0 \)
2. \( 29 - 5D + 2E + F = 0 \)
3. \( 45 + 3D - 6E + F = 0 \)

Let's solve this step-by-step. I'll continue by eliminating \(F\).The solution to the system of equations is:

- \( D = -4 \)
- \( E = -2 \)
- \( F = -45 \)

### Step 3: Writing the equation of the circle

Substituting these values into the general equation of a circle:

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

we get:

\[
x^2 + y^2 - 4x - 2y - 45 = 0
\]

Thus, the equation of the circle is:

\[
x^2 + y^2 - 4x - 2y - 45 = 0
\]

Would you like a detailed explanation of any part of this solution? Here are some related questions for further exploration:

1. How do you find the radius of the circle given the equation?
2. Can we convert this equation into center-radius form?
3. How would we determine if a point lies inside the circle?
4. What is the geometric significance of the constants \(D\) and \(E\)?
5. How do you generalize this method for any three points?

**Tip:** The general form of a circle equation \(x^2 + y^2 + Dx + Ey + F = 0\) can be easily converted into center-radius form to identify the center and radius directly.

Find the equation of the circle passing through (-3,6), (-5,2), and (3,-6).

Discover how to find the equation of a circle passing through the points (-3,6), (-5,2), and (3,-6). This detailed solution covers the general equation of a circle, solving a system of equations, and finding the constants D, E, and F. Suitable for students in Grades 10-12.

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