Certainly! Let's solve Question 54 from the image:

**Question 54:**
What is the minimum distance of the circle given by the equation \( x^2 + 16x + y^2 + 12y - 5 = 0 \) from the coordinate axes?

**Solution:**

First, we need to rewrite the equation of the circle in its standard form by completing the square.

1. **Group the \(x\) and \(y\) terms:**

   \[
   (x^2 + 16x) + (y^2 + 12y) = 5
   \]

2. **Complete the square for both \(x\) and \(y\):**

   For \(x^2 + 16x\):
   \[
   x^2 + 16x = (x + 8)^2 - 64
   \]

   For \(y^2 + 12y\):
   \[
   y^2 + 12y = (y + 6)^2 - 36
   \]

   Substitute these back into the equation:
   \[
   (x + 8)^2 - 64 + (y + 6)^2 - 36 = 5
   \]

3. **Simplify the equation:**

   \[
   (x + 8)^2 + (y + 6)^2 = 105
   \]

   So, the equation of the circle in standard form is:
   \[
   (x + 8)^2 + (y + 6)^2 = 105
   \]

   This represents a circle with center \((-8, -6)\) and radius \( \sqrt{105} \).

4. **Calculate the distances from the center to the x-axis and y-axis:**

   - Distance from the center \((-8, -6)\) to the x-axis: \( |y| = |-6| = 6 \)
   - Distance from the center to the y-axis: \( |x| = |-8| = 8 \)

5. **Determine the minimum distance from the circle to the axes:**

   The minimum distance from the circle to the x-axis or y-axis is the distance from the center to the axis minus the radius:
   - Distance to the x-axis: \( 6 - \sqrt{105} \)
   - Distance to the y-axis: \( 8 - \sqrt{105} \)

   Since \( 6 < 8 \), the minimum distance to the x-axis is smaller.

   Therefore, the minimum distance is \( 6 - \sqrt{105} \).

**Answer:** The minimum distance from the circle to the coordinate axes is \( 6 - \sqrt{105} \), which corresponds to **Option (B)**.

Would you like further details or have any other questions? 

Here are 5 related questions that might interest you:
1. How do you derive the general form of a circle from its standard form?
2. What is the significance of the center and radius in a circle's equation?
3. How do you calculate the distance between two points on a coordinate plane?
4. What are the applications of completing the square in other areas of mathematics?
5. How do you find the distance between a point and a line?

**Tip:** Understanding how to complete the square is essential for solving many quadratic-related problems, including circles and parabolas.

Learn how to calculate the minimum distance of a circle from the coordinate axes using the circle's equation and distance formulas. This problem involves completing the square and understanding circle equations in coordinate geometry. Suitable for high school students.

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