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To solve this problem, we need to first rewrite the given equation of the circle into its standard form and then find the minimum distance from the circle to the coordinate axes.

### Step 1: Write the equation in standard form.
The given equation is:

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

We need to complete the square for both \(x\) and \(y\) terms.

#### Completing the square for \(x\):
Start with \(x^2 - 10x\). To complete the square, take half of \(-10\), square it, and add it:

\[
\left(\frac{-10}{2}\right)^2 = 25
\]

So, \(x^2 - 10x + 25\) is a perfect square:

\[
(x - 5)^2
\]

#### Completing the square for \(y\):
Now for \(y^2 + 12y\), take half of \(12\), square it, and add it:

\[
\left(\frac{12}{2}\right)^2 = 36
\]

So, \(y^2 + 12y + 36\) is a perfect square:

\[
(y + 6)^2
\]

### Step 2: Adjust the equation.
We added \(25\) and \(36\) to the equation, so to maintain equality, we subtract \(25 + 36 = 61\) from the right side:

\[
x^2 - 10x + y^2 + 12y + 57 = (x - 5)^2 + (y + 6)^2 - 61 + 57 = (x - 5)^2 + (y + 6)^2 - 4 = 0
\]

Thus, the equation becomes:

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

This is the equation of a circle with center \((5, -6)\) and radius \(2\) (since \(4 = 2^2\)).

### Step 3: Find the minimum distance from the coordinate axes.
The minimum distance from the circle to the coordinate axes is the shortest distance from the center \((5, -6)\) to either the \(x\)-axis or the \(y\)-axis, minus the radius of the circle.

- The distance from the center to the \(y\)-axis (i.e., the vertical distance) is \(5\).
- The distance from the center to the \(x\)-axis (i.e., the horizontal distance) is \(|-6| = 6\).

The minimum of these two distances is \(5\). Since the radius of the circle is \(2\), we subtract the radius from the distance to get the minimum distance from the circle to the axes:

\[
\text{Minimum distance} = 5 - 2 = 3
\]

Thus, the minimum distance from the circle to the coordinate axes is **3**, which corresponds to option **A**.

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Would you like a detailed explanation of any step? Here are some related questions to expand on this concept:

1. How do you complete the square for a quadratic equation?
2. What is the general formula for the distance from a point to an axis?
3. How do you find the radius of a circle from its equation?
4. What is the significance of the distance from the center of the circle to the coordinate axes?
5. How does completing the square help in converting equations of conics to standard form?

**Tip:** Always check if completing the square is necessary to rewrite a quadratic equation into a more useful form, such as the standard form of a circle.

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

Learn to find the minimum distance from the circle with the equation x^2 - 10x + y^2 + 12y + 57 = 0 to the coordinate axes by converting it to the standard form and calculating the distance. This problem involves algebraic manipulation, circle equations, and geometry concepts.

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