To find the equation of the circle that passes through the points (6, 5) and (4, 1) with its center lying on the line given by the equation \(4x + y - 16 = 0\), we can follow these steps:
### Step 1: Define the center of the circle
Let the center of the circle be \((h, k)\). The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
### Step 2: Use the points on the circle
Since the circle passes through the point (6, 5), substituting these coordinates into the circle's equation gives:
\[
(6 - h)^2 + (5 - k)^2 = r^2 \quad \text{(1)}
\]
Similarly, since the circle also passes through the point (4, 1), substituting these coordinates gives:
\[
(4 - h)^2 + (1 - k)^2 = r^2 \quad \text{(2)}
\]
### Step 3: Set the equations equal to each other
Since both equations equal \(r^2\), we can set them equal to each other:
\[
(6 - h)^2 + (5 - k)^2 = (4 - h)^2 + (1 - k)^2
\]
### Step 4: Expand both sides
Expanding both sides:
- Left side:
\[
(6 - h)^2 = 36 - 12h + h^2
\]
\[
(5 - k)^2 = 25 - 10k + k^2
\]
Thus, the left side becomes:
\[
36 - 12h + h^2 + 25 - 10k + k^2 = h^2 - 12h + k^2 - 10k + 61
\]
- Right side:
\[
(4 - h)^2 = 16 - 8h + h^2
\]
\[
(1 - k)^2 = 1 - 2k + k^2
\]
Thus, the right side becomes:
\[
16 - 8h + h^2 + 1 - 2k + k^2 = h^2 - 8h + k^2 - 2k + 17
\]
### Step 5: Set the expanded forms equal
Now, we can set the expanded forms equal:
\[
h^2 - 12h + k^2 - 10k + 61 = h^2 - 8h + k^2 - 2k + 17
\]
### Step 6: Simplify the equation
Cancel \(h^2\) and \(k^2\) from both sides:
\[
-12h - 10k + 61 = -8h - 2k + 17
\]
Rearranging gives:
\[
-12h + 8h - 10k + 2k + 61 - 17 = 0
\]
This simplifies to:
\[
-4h - 8k + 44 = 0
\]
Dividing through by -4 gives:
\[
h + 2k = 11 \quad \text{(3)}
\]
### Step 7: Use the line equation
The center \((h, k)\) lies on the line given by \(4h + k - 16 = 0\). Rearranging gives:
\[
4h + k = 16 \quad \text{(4)}
\]
### Step 8: Solve the system of equations
Now we have two equations:
1. \(h + 2k = 11\) (from step 6)
2. \(4h + k = 16\) (from step 7)
We can solve these equations simultaneously. From equation (3), we can express \(k\) in terms of \(h\):
\[
k = \frac{11 - h}{2}
\]
Substituting this into equation (4):
\[
4h + \frac{11 - h}{2} = 16
\]
Multiplying through by 2 to eliminate the fraction:
\[
8h + 11 - h = 32
\]
This simplifies to:
\[
7h + 11 = 32
\]
So,
\[
7h = 21 \quad \Rightarrow \quad h = 3
\]
### Step 9: Find \(k\)
Substituting \(h = 3\) back into equation (3):
\[
3 + 2k = 11 \quad \Rightarrow \quad 2k = 8 \quad \Rightarrow \quad k = 4
\]
### Step 10: Find the radius \(r\)
Now we have the center of the circle \((h, k) = (3, 4)\). To find the radius \(r\), we can substitute \(h\) and \(k\) back into either equation (1) or (2). Using equation (1):
\[
(6 - 3)^2 + (5 - 4)^2 = r^2
\]
Calculating gives:
\[
3^2 + 1^2 = r^2 \quad \Rightarrow \quad 9 + 1 = r^2 \quad \Rightarrow \quad r^2 = 10
\]
### Step 11: Write the equation of the circle
The equation of the circle is:
\[
(x - 3)^2 + (y - 4)^2 = 10
\]
### Step 12: Expand the equation
Expanding this gives:
\[
(x^2 - 6x + 9) + (y^2 - 8y + 16) = 10
\]
Combining terms results in:
\[
x^2 + y^2 - 6x - 8y + 25 - 10 = 0
\]
Thus, the final equation of the circle is:
\[
x^2 + y^2 - 6x - 8y + 15 = 0
\]
