find the equation circle which is passes through the points `(4,1),(6,5)` and centre lies on `4x+y=16` is

Answer

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

### Step 1: Write the 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 the points into the circle's equation
We have two points through which the circle passes: (4, 1) and (6, 5). We can substitute these points into the circle's equation to create two equations.

For the point (4, 1):
\[ (4 - h)^2 + (1 - k)^2 = r^2 \tag{1} \]

For the point (6, 5):
\[ (6 - h)^2 + (5 - k)^2 = r^2 \tag{2} \]

### Step 3: Expand both equations
Expanding equation (1):
\[ (4 - h)^2 + (1 - k)^2 = r^2 \]
\[ 16 - 8h + h^2 + 1 - 2k + k^2 = r^2 \]
\[ h^2 + k^2 - 8h - 2k + 17 = r^2 \tag{3} \]

Expanding equation (2):
\[ (6 - h)^2 + (5 - k)^2 = r^2 \]
\[ 36 - 12h + h^2 + 25 - 10k + k^2 = r^2 \]
\[ h^2 + k^2 - 12h - 10k + 61 = r^2 \tag{4} \]

### Step 4: Set equations (3) and (4) equal to each other
Since both equations are equal to \(r^2\), 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
Cancelling \(h^2\) and \(k^2\) from both sides:
\[ -8h - 2k + 17 = -12h - 10k + 61 \]
Rearranging gives:
\[ 4h + 8k = 44 \]
Dividing by 4:
\[ h + 2k = 11 \tag{5} \]

### Step 6: Use the center condition
The center of the circle lies on the line \(4x + y = 16\). Substituting \(h\) and \(k\):
\[ 4h + k = 16 \tag{6} \]

### Step 7: Solve the system of equations
Now we have two equations (5) and (6):
1. \(h + 2k = 11\)
2. \(4h + k = 16\)

From equation (5), express \(k\) in terms of \(h\):
\[ k = \frac{11 - h}{2} \]

Substituting into equation (6):
\[ 4h + \frac{11 - h}{2} = 16 \]
Multiplying through by 2 to eliminate the fraction:
\[ 8h + 11 - h = 32 \]
\[ 7h = 21 \implies h = 3 \]

### Step 8: Find \(k\)
Substituting \(h = 3\) back into equation (5):
\[ 3 + 2k = 11 \implies 2k = 8 \implies k = 4 \]

### Step 9: Find the radius \(r\)
Now substitute \(h = 3\) and \(k = 4\) into either equation (1) or (2) to find \(r\). Using equation (1):
\[ (4 - 3)^2 + (1 - 4)^2 = r^2 \]
\[ 1 + 9 = r^2 \implies r^2 = 10 \implies r = \sqrt{10} \]

### Step 10: Write the equation of the circle
Now we have \(h = 3\), \(k = 4\), and \(r^2 = 10\). The equation of the circle is:
\[ (x - 3)^2 + (y - 4)^2 = 10 \]

### Final Equation
Expanding this gives:
\[ x^2 - 6x + 9 + y^2 - 8y + 16 = 10 \]
\[ x^2 + y^2 - 6x - 8y + 15 = 0 \]

### Conclusion
The equation of the circle is:
\[ x^2 + y^2 - 6x - 8y + 15 = 0 \]