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

Answer

To find the equation of the circle that passes through the points (4, 1) and (6, 5) and whose center lies on the line \(4x + y = 16\), we can follow these steps:

### Step 1: General Equation of the Circle
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: Substitute Point (4, 1)
Since the circle passes through the point (4, 1), we substitute \(x = 4\) and \(y = 1\) into the circle's equation:

\[ (4 - h)^2 + (1 - k)^2 = r^2 \]

Expanding this gives:

\[ (4 - h)^2 + (1 - k)^2 = r^2 \]
\[ (16 - 8h + h^2) + (1 - 2k + k^2) = r^2 \]
\[ h^2 + k^2 - 8h - 2k + 17 = r^2 \quad \text{(Equation 1)} \]

### Step 3: Substitute Point (6, 5)
Now, substitute the second point (6, 5) into the circle's equation:

\[ (6 - h)^2 + (5 - k)^2 = r^2 \]

Expanding this gives:

\[ (6 - h)^2 + (5 - k)^2 = r^2 \]
\[ (36 - 12h + h^2) + (25 - 10k + k^2) = r^2 \]
\[ h^2 + k^2 - 12h - 10k + 61 = r^2 \quad \text{(Equation 2)} \]

### Step 4: Set Equations Equal
Since both equations equal \(r^2\), we can set them equal to each other:

\[ h^2 + k^2 - 8h - 2k + 17 = h^2 + k^2 - 12h - 10k + 61 \]

### Step 5: Simplify the Equation
Cancel \(h^2\) and \(k^2\) from both sides:

\[ -8h - 2k + 17 = -12h - 10k + 61 \]

Rearranging gives:

\[ 4h + 8k - 44 = 0 \]

### Step 6: Center Lies on the Line
Since the center \((h, k)\) lies on the line \(4x + y = 16\), we have:

\[ 4h + k = 16 \quad \text{(Equation 3)} \]

### Step 7: Solve the System of Equations
Now we have two equations:

1. \(4h + 8k = 44\)
2. \(4h + k = 16\)

From Equation 3, we can express \(k\) in terms of \(h\):

\[ k = 16 - 4h \]

Substituting this into Equation 1:

\[ 4h + 8(16 - 4h) = 44 \]
\[ 4h + 128 - 32h = 44 \]
\[ -28h + 128 = 44 \]
\[ -28h = 44 - 128 \]
\[ -28h = -84 \]
\[ h = 3 \]

### Step 8: Find \(k\)
Substituting \(h = 3\) back into Equation 3 to find \(k\):

\[ k = 16 - 4(3) = 16 - 12 = 4 \]

### Step 9: Find \(r^2\)
Now substitute \(h = 3\) and \(k = 4\) back into either Equation 1 or Equation 2 to find \(r^2\). Using Equation 1:

\[ 3^2 + 4^2 - 8(3) - 2(4) + 17 = r^2 \]
\[ 9 + 16 - 24 - 8 + 17 = r^2 \]
\[ 10 = r^2 \]

### Step 10: Write the Circle's Equation
Now we have \(h = 3\), \(k = 4\), and \(r^2 = 10\). The equation of the circle is:

\[ (x - 3)^2 + (y - 4)^2 = 10 \]

Expanding this gives:

\[ x^2 - 6x + 9 + y^2 - 8y + 16 = 10 \]
\[ x^2 + y^2 - 6x - 8y + 15 = 0 \]

### Final Answer
The equation of the circle is:

\[ x^2 + y^2 - 6x - 8y + 15 = 0 \]