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To determine the equation of a circle, we need to use the general form for a circle centered at \((h, k)\) with radius \(r\):

\[
(x - h)^2 + (y - k)^2 = r^2
\]

### Given Information
- Center of the circle: \((h, k) = (2, -3)\)
- The circle passes through the point \((4, 1)\).

### Step-by-Step Solution

1. **Set up the equation with the center**:
   Substituting \(h = 2\) and \(k = -3\) into the general equation, we get:
   \[
   (x - 2)^2 + (y + 3)^2 = r^2
   \]

2. **Calculate the radius**:
   Since the circle passes through the point \((4, 1)\), we can use this point to find \(r\).
   
   Substitute \((x, y) = (4, 1)\) into the equation:
   \[
   (4 - 2)^2 + (1 + 3)^2 = r^2
   \]
   Simplifying:
   \[
   (2)^2 + (4)^2 = r^2
   \]
   \[
   4 + 16 = r^2
   \]
   \[
   r^2 = 20
   \]

3. **Write the final equation**:
   Substitute \(r^2 = 20\) back into the circle equation:
   \[
   (x - 2)^2 + (y + 3)^2 = 20
   \]

### Final Answer
The equation of the circle is:
\[
(x - 2)^2 + (y + 3)^2 = 20
\]

Would you like more details on any specific part of this solution, or do you have further questions?

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### Related Questions
1. How can we find the radius of a circle given two points on the circumference?
2. What is the distance formula used for in geometry, and how is it applied to circles?
3. How can you determine if a given point lies on, inside, or outside a circle?
4. What other forms can the equation of a circle take, and when are they used?
5. How do you convert the general form of a circle equation to the standard form?

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### Tip
When finding the equation of a circle given two points, always check if one point is the center to simplify the radius calculation using the distance formula.

Nosaki vienādojumu riņķa līnijai, kuras centrs atrodas punktā (2, -3) un kura iet caur punktu (4, 1).

Find the equation of a circle centered at (2, -3) that passes through the point (4, 1). This problem involves calculating the radius using the distance formula and applying the standard equation for a circle. Suitable for high school students.

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