Find the equation of circle passing through the points (0,5) and (6,1) and whose centre lies on the line 2x+5y=25.

Answer

To find the equation of the circle passing through the points (0, 5) and (6, 1) with its center lying on the line \(2x + 5y = 25\), 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 (0, 5)
Substituting the point (0, 5) into the general equation:
\[ 0^2 + 5^2 + 2g(0) + 2f(5) + c = 0 \]
This simplifies to:
\[ 25 + 10f + c = 0 \quad \text{(Equation 1)} \]

### Step 3: Substitute the Second Point (6, 1)
Now, substitute the point (6, 1) into the general equation:
\[ 6^2 + 1^2 + 2g(6) + 2f(1) + c = 0 \]
This simplifies to:
\[ 36 + 1 + 12g + 2f + c = 0 \]
or
\[ 37 + 12g + 2f + c = 0 \quad \text{(Equation 2)} \]

### Step 4: Center Lies on the Given Line
The center of the circle \((-g, -f)\) lies on the line \(2x + 5y = 25\). Substituting \(-g\) and \(-f\) into the line equation gives:
\[ 2(-g) + 5(-f) = 25 \]
which simplifies to:
\[ -2g - 5f = 25 \quad \text{(Equation 3)} \]

### Step 5: Solve the System of Equations
Now we have three equations:
1. \(10f + c = -25\) (from Equation 1)
2. \(12g + 2f + c = -37\) (from Equation 2)
3. \(-2g - 5f = 25\) (from Equation 3)

### Step 6: Eliminate \(c\)
From Equation 1, we can express \(c\) as:
\[ c = -25 - 10f \]
Substituting this into Equation 2:
\[ 12g + 2f - 25 - 10f = -37 \]
This simplifies to:
\[ 12g - 8f - 12 = 0 \quad \text{(Equation 4)} \]

### Step 7: Solve for \(f\) and \(g\)
Now we will solve Equations 3 and 4 together. From Equation 3:
\[ -2g - 5f = 25 \implies 2g + 5f = -25 \]
Now, we can add Equation 4 and this modified Equation 3:
\[ (12g - 8f) + (2g + 5f) = 12 - 25 \]
This simplifies to:
\[ 14g - 3f = -13 \quad \text{(Equation 5)} \]

### Step 8: Solve for \(f\)
Now, we can express \(g\) in terms of \(f\) from Equation 5:
\[ g = \frac{-13 + 3f}{14} \]

### Step 9: Substitute \(g\) into Equation 3
Substituting \(g\) back into Equation 3:
\[ -2\left(\frac{-13 + 3f}{14}\right) - 5f = 25 \]
This leads to:
\[ \frac{26 - 6f}{14} - 5f = 25 \]
Multiplying through by 14 to eliminate the fraction:
\[ 26 - 6f - 70f = 350 \]
Combining like terms gives:
\[ -76f = 324 \implies f = -\frac{27}{19} \]

### Step 10: Substitute \(f\) back to find \(g\)
Now substitute \(f\) back into the equation for \(g\):
\[ g = \frac{-13 + 3\left(-\frac{27}{19}\right)}{14} = \frac{-13 - \frac{81}{19}}{14} = \frac{-\frac{247}{19}}{14} = -\frac{247}{266} \]

### Step 11: Find \(c\)
Now substitute \(f\) back into Equation 1 to find \(c\):
\[ c = -25 - 10\left(-\frac{27}{19}\right) = -25 + \frac{270}{19} = -\frac{475}{19} \]

### Step 12: Write the Circle Equation
Now we can write the equation of the circle:
\[ x^2 + y^2 + 2\left(-\frac{247}{266}\right)x + 2\left(-\frac{27}{19}\right)y + \left(-\frac{475}{19}\right) = 0 \]

### Final Circle Equation
After simplifying, we arrive at the final equation of the circle.