Find the equation of the circle which passes through the points (5,0) and (1,4) and whose centre lies on the line x + y - 3 = 0.

Answer

To find the equation of the circle that passes through the points (5, 0) and (1, 4) with its center lying on the line \(x + y - 3 = 0\), we can follow these steps:

### Step 1: Define the center of the circle
Let the center of the circle be \(C(h, k)\). Since the center lies on the line \(x + y - 3 = 0\), we can express \(k\) in terms of \(h\):
\[ k = 3 - h \]

### Step 2: Use the distance formula
The radius of the circle is the distance from the center \(C(h, k)\) to the points (5, 0) and (1, 4). We can set up two equations based on the distance formula.

1. Distance from \(C(h, k)\) to (5, 0):
\[ \sqrt{(h - 5)^2 + (k - 0)^2} \]
2. Distance from \(C(h, k)\) to (1, 4):
\[ \sqrt{(h - 1)^2 + (k - 4)^2} \]

Since both distances represent the radius, we can set them equal to each other:
\[ \sqrt{(h - 5)^2 + (k - 0)^2} = \sqrt{(h - 1)^2 + (k - 4)^2} \]

### Step 3: Square both sides
Squaring both sides to eliminate the square roots gives:
\[ (h - 5)^2 + k^2 = (h - 1)^2 + (k - 4)^2 \]

### Step 4: Expand both sides
Expanding both sides:
\[ (h^2 - 10h + 25 + k^2) = (h^2 - 2h + 1 + k^2 - 8k + 16) \]

### Step 5: Simplify the equation
Cancelling \(h^2\) and \(k^2\) from both sides:
\[ -10h + 25 = -2h + 17 - 8k \]
Rearranging gives:
\[ -10h + 2h + 8k = 17 - 25 \]
\[ -8h + 8k = -8 \]
Dividing through by 8:
\[ -k + h = 1 \quad \text{or} \quad k = h - 1 \]

### Step 6: Substitute back for k
Now we have two expressions for \(k\):
1. \(k = 3 - h\)
2. \(k = h - 1\)

Setting them equal to each other:
\[ 3 - h = h - 1 \]
Solving for \(h\):
\[ 3 + 1 = 2h \quad \Rightarrow \quad 4 = 2h \quad \Rightarrow \quad h = 2 \]

### Step 7: Find k
Substituting \(h = 2\) back into \(k = 3 - h\):
\[ k = 3 - 2 = 1 \]

### Step 8: Find the radius
Now we have the center of the circle \(C(2, 1)\). To find the radius, we can use the distance from the center to one of the points, say (5, 0):
\[ r = \sqrt{(2 - 5)^2 + (1 - 0)^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

### Step 9: Write the equation of the circle
The standard form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 2\), \(k = 1\), and \(r^2 = 10\):
\[ (x - 2)^2 + (y - 1)^2 = 10 \]

### Final Equation
Thus, the equation of the circle is:
\[ (x - 2)^2 + (y - 1)^2 = 10 \]