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The given problem asks to find the equation of a circle that:

- Passes through the points \( (4, 1) \) and \( (6, 5) \)
- Has its center on the straight line \( 4x + y = 16 \).

### Steps to solve the problem:

1. **General Equation of the Circle:**
   The general equation of a circle is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   where \( (h, k) \) is the center and \( r \) is the radius.

2. **Conditions on the Circle:**
   - Since the circle passes through two points, we will substitute these points into the general equation to get two conditions involving \( h \), \( k \), and \( r^2 \).
   - The center \( (h, k) \) lies on the line \( 4x + y = 16 \). This will give us a third equation relating \( h \) and \( k \).

3. **Substitute the Points into the Equation:**
   First, substitute the points \( (4, 1) \) and \( (6, 5) \) into the equation of the circle.

   - For point \( (4, 1) \):
     \[
     (4 - h)^2 + (1 - k)^2 = r^2
     \]
     Expanding this:
     \[
     (16 - 8h + h^2) + (1 - 2k + k^2) = r^2
     \]
     Simplified:
     \[
     h^2 + k^2 - 8h - 2k + 17 = r^2 \tag{1}
     \]

   - For point \( (6, 5) \):
     \[
     (6 - h)^2 + (5 - k)^2 = r^2
     \]
     Expanding this:
     \[
     (36 - 12h + h^2) + (25 - 10k + k^2) = r^2
     \]
     Simplified:
     \[
     h^2 + k^2 - 12h - 10k + 61 = r^2 \tag{2}
     \]

4. **Substitute the Center's Condition:**
   Since the center \( (h, k) \) lies on the line \( 4x + y = 16 \), we substitute \( (h, k) \) into this line equation:
   \[
   4h + k = 16 \tag{3}
   \]

5. **Solving the System of Equations:**
   Now we solve the system formed by equations (1), (2), and (3).

Let me compute the solution.The solution to the system of equations gives us the center and radius of the circle:

- The center of the circle is \( (h, k) = (3, 4) \)
- The radius squared is \( r^2 = 10 \)

### Equation of the Circle:
Now, we can write the equation of the circle using the general form \( (x - h)^2 + (y - k)^2 = r^2 \).

Substituting \( h = 3 \), \( k = 4 \), and \( r^2 = 10 \), we get:
\[
(x - 3)^2 + (y - 4)^2 = 10
\]

This is the required equation of the circle.

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Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How do you find the radius of the circle from the given points?
2. What is the method for finding the equation of a circle passing through three points?
3. How would you solve for the center if the circle passes through different points?
4. Can a circle have its center on a curved path instead of a straight line?
5. What happens if the given line does not intersect the circle's center?

**Tip:** When solving systems of equations involving circles, always check if substituting conditions such as points and lines simplifies the problem effectively.

Find the equation of the circle passing through points (4, 1) and (6, 5) and whose center is on the straight line 4x + y = 16.

This problem explores how to find the equation of a circle that passes through points (4, 1) and (6, 5), with its center lying on the line 4x + y = 16. Learn how to apply circle equations and solve systems of equations to find the circle's center and radius.

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