To find the equation of a circle passing through three points, we 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. Since the circle passes through the points $(-3,6)$, $(-5,2)$, and $(3,-6)$, we substitute these coordinates into the equation to form a system of equations.

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

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

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

Now we solve this system of equations to find $D$, $E$, and $F$.

System of Equations:

1. $-3D + 6E + F = -45$
2. $-5D + 2E + F = -29$
3. $3D - 6E + F = -45$

Let's solve these equations.

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Here are some related questions you may find helpful:

1. How do you derive the equation of a circle given three points in general?
2. What are other methods to solve systems of equations?
3. Can you explain the geometric interpretation of a circle equation?
4. What role do determinants play in circle equations?
5. How can you use matrices to solve this problem efficiently?

Tip: Always double-check your algebraic manipulation when working with multiple variables to avoid calculation errors.