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The problem asks for the minimum distance of a circle, described by the equation:

\[
x^2 - 10x + y^2 + 12y + 57 = 0
\]

from the coordinate axes.

### Step 1: Rewrite the circle's equation in standard form
To identify the center and radius of the circle, we need to complete the square for both \(x\) and \(y\) terms.

#### Completing the square for \(x\):
- Start with the \(x\)-terms: \(x^2 - 10x\).
- To complete the square: \( (x - 5)^2 = x^2 - 10x + 25 \).

Add 25 and subtract 25 to keep the equation balanced.

#### Completing the square for \(y\):
- Now, for the \(y\)-terms: \(y^2 + 12y\).
- To complete the square: \( (y + 6)^2 = y^2 + 12y + 36 \).

Add 36 and subtract 36 to keep the equation balanced.

### Step 2: Form the standard equation
Now, rewrite the equation by including the completed squares:

\[
(x - 5)^2 - 25 + (y + 6)^2 - 36 + 57 = 0
\]

Simplify:

\[
(x - 5)^2 + (y + 6)^2 - 4 = 0
\]

\[
(x - 5)^2 + (y + 6)^2 = 4
\]

This is the standard form of a circle with center \((5, -6)\) and radius \(r = 2\).

### Step 3: Find the minimum distance to the coordinate axes
The center of the circle is at \((5, -6)\), and the radius is 2. The minimum distance from the circle to the coordinate axes occurs when the center's distances from the axes are reduced by the radius.

- Distance from the center to the \(y\)-axis (at \(x = 0\)): \(5\).
- Distance from the center to the \(x\)-axis (at \(y = 0\)): \(|-6| = 6\).

The minimum of these distances minus the radius is:

- Distance to the \(y\)-axis: \(5 - 2 = 3\).
- Distance to the \(x\)-axis: \(6 - 2 = 4\).

Thus, the minimum distance from the circle to the coordinate axes is **3**.

### Answer:
The correct choice is **A) 3**.

---

Would you like further details or have any other questions? Here are five related questions for you to explore:

1. How do you complete the square for more complex quadratic expressions?
2. What is the geometric interpretation of the center and radius of a circle in relation to the coordinate plane?
3. How do you calculate the distance from a point to a line (like the \(x\)- or \(y\)-axis)?
4. What are the different types of transformations that can be applied to conic sections like circles?
5. How can we generalize the approach to find the minimum distance from any curve to the coordinate axes?

**Tip:** Completing the square is a powerful tool for converting quadratic equations into a more usable form, especially for identifying geometric properties like centers and radii.

What is the minimum distance of the circle given by the equation x^2 - 10x + y^2 + 12y + 57 = 0 from the coordinate axes?

Solve for the minimum distance of the circle described by x^2 - 10x + y^2 + 12y + 57 = 0 from the coordinate axes. This problem involves completing the square, understanding circle properties, and using geometric distances.

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