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 expressed 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)
Since the circle passes through the point (0, 5), we can substitute \(x = 0\) and \(y = 5\) into the circle's equation:
\[ 0^2 + 5^2 + 2g(0) + 2f(5) + c = 0 \]
This simplifies to:
\[ 25 + 10f + c = 0 \]
Rearranging gives us:
\[ 10f + c = -25 \quad \text{(Equation 1)} \]

### Step 3: Substitute the Second Point (6, 1)
Next, we substitute the second point (6, 1) into the circle's equation:
\[ 6^2 + 1^2 + 2g(6) + 2f(1) + c = 0 \]
This simplifies to:
\[ 36 + 1 + 12g + 2f + c = 0 \]
or:
\[ 37 + 12g + 2f + c = 0 \]
Rearranging gives us:
\[ 12g + 2f + c = -37 \quad \text{(Equation 2)} \]

### Step 4: Center Lies on the Line
The center of the circle is \((-g, -f)\). Since it lies on the line \(2x + 5y = 25\), we substitute \(-g\) for \(x\) and \(-f\) for \(y\):
\[ 2(-g) + 5(-f) = 25 \]
This simplifies to:
\[ -2g - 5f = 25 \]
or:
\[ 2g + 5f = -25 \quad \text{(Equation 3)} \]

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

We can express \(c\) from Equation 1:
\[ c = -25 - 10f \]

Substituting \(c\) into Equation 2:
\[ 12g + 2f - 25 - 10f = -37 \]
This simplifies to:
\[ 12g - 8f = -12 \]
or:
\[ 3g - 2f = -3 \quad \text{(Equation 4)} \]

Now we have:
1. \(2g + 5f = -25\) (Equation 3)
2. \(3g - 2f = -3\) (Equation 4)

### Step 6: Solve Equations 3 and 4
From Equation 4, we can express \(g\):
\[ 3g = -3 + 2f \implies g = \frac{-3 + 2f}{3} \]

Substituting \(g\) into Equation 3:
\[ 2\left(\frac{-3 + 2f}{3}\right) + 5f = -25 \]
Multiplying through by 3 to eliminate the fraction:
\[ 2(-3 + 2f) + 15f = -75 \]
This simplifies to:
\[ -6 + 4f + 15f = -75 \]
Combining like terms:
\[ 19f = -69 \implies f = -\frac{69}{19} \]

### Step 7: Find \(g\) and \(c\)
Substituting \(f\) back into the equation for \(g\):
\[ g = \frac{-3 + 2\left(-\frac{69}{19}\right)}{3} = \frac{-3 - \frac{138}{19}}{3} = \frac{-\frac{57}{19} - \frac{138}{19}}{3} = \frac{-195/19}{3} = -\frac{65}{19} \]

Now substituting \(f\) into Equation 1 to find \(c\):
\[ c = -25 - 10\left(-\frac{69}{19}\right) = -25 + \frac{690}{19} = \frac{-475 + 690}{19} = \frac{215}{19} \]

### Step 8: Final Equation of the Circle
Now substituting \(g\), \(f\), and \(c\) back into the general equation:
\[ x^2 + y^2 + 2\left(-\frac{65}{19}\right)x + 2\left(-\frac{69}{19}\right)y + \frac{215}{19} = 0 \]
Multiplying through by 19 to eliminate the fractions:
\[ 19x^2 + 19y^2 - 130x - 138y + 215 = 0 \]

### Final Answer
The equation of the circle is:
\[ 19x^2 + 19y^2 - 130x - 138y + 215 = 0 \]