Find the equation of circle passing through the point (2,3) and (-1,1) and whose centre is on the line x - 3y - 11 = 0

Answer

To find the equation of the circle that passes through the points (2, 3) and (-1, 1), and whose center lies on the line \( x - 3y - 11 = 0 \), we can follow these steps:

### Step 1: General Equation of the Circle
The general equation of a circle can be written as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \((-g, -f)\) is the center of the circle.

### Step 2: Substitute the First Point (2, 3)
Since the circle passes through the point (2, 3), we substitute \(x = 2\) and \(y = 3\) into the circle's equation:
\[ (2)^2 + (3)^2 + 2g(2) + 2f(3) + c = 0 \]
Calculating this gives:
\[ 4 + 9 + 4g + 6f + c = 0 \implies 4g + 6f + c + 13 = 0 \implies 4g + 6f + c = -13 \quad \text{(Equation 1)} \]

### Step 3: Substitute the Second Point (-1, 1)
Now, we substitute the second point (-1, 1) into the circle's equation:
\[ (-1)^2 + (1)^2 + 2g(-1) + 2f(1) + c = 0 \]
Calculating this gives:
\[ 1 + 1 - 2g + 2f + c = 0 \implies -2g + 2f + c + 2 = 0 \implies -2g + 2f + c = -2 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We now have two equations:
1. \(4g + 6f + c = -13\) (Equation 1)
2. \(-2g + 2f + c = -2\) (Equation 2)

We can eliminate \(c\) by subtracting Equation 2 from Equation 1:
\[ (4g + 6f + c) - (-2g + 2f + c) = -13 - (-2) \]
This simplifies to:
\[ 6g + 4f = -11 \quad \text{(Equation 3)} \]

### Step 5: Center on the Given Line
The center of the circle \((-g, -f)\) lies on the line \(x - 3y - 11 = 0\). Substituting \((-g, -f)\) into the line equation gives:
\[ -g - 3(-f) - 11 = 0 \implies -g + 3f = 11 \quad \text{(Equation 4)} \]

### Step 6: Solve Equations 3 and 4
Now we have two equations:
1. \(6g + 4f = -11\) (Equation 3)
2. \(-g + 3f = 11\) (Equation 4)

From Equation 4, we can express \(g\) in terms of \(f\):
\[ g = 3f - 11 \]
Substituting this into Equation 3:
\[ 6(3f - 11) + 4f = -11 \]
Expanding this gives:
\[ 18f - 66 + 4f = -11 \implies 22f - 66 = -11 \implies 22f = 55 \implies f = \frac{55}{22} = \frac{5}{2} \]

### Step 7: Find \(g\)
Now substitute \(f\) back to find \(g\):
\[ g = 3\left(\frac{5}{2}\right) - 11 = \frac{15}{2} - 11 = \frac{15}{2} - \frac{22}{2} = -\frac{7}{2} \]

### Step 8: Find \(c\)
Now substitute \(g\) and \(f\) back into Equation 1 to find \(c\):
\[ 4\left(-\frac{7}{2}\right) + 6\left(\frac{5}{2}\right) + c = -13 \]
Calculating this gives:
\[ -14 + 15 + c = -13 \implies c = -13 + 14 - 15 = -14 \]

### Step 9: Write the Final Equation
Now we have \(g = -\frac{7}{2}\), \(f = \frac{5}{2}\), and \(c = -14\). The equation of the circle is:
\[ x^2 + y^2 - 7x + 5y - 14 = 0 \]

### Final Answer
The equation of the required circle is:
\[ x^2 + y^2 - 7x + 5y - 14 = 0 \]