Find the equation of the circle which passes through the points (1,-2) and (4,-3) and which has its centre on the straight line 3x+4y=5.

Answer

To find the equation of the circle that passes through the points (1, -2) and (4, -3) and has its center on the line \(3x + 4y = 5\), we can follow these steps:

### Step 1: Let the center of the circle be \((h, k)\)
Since the center of the circle lies on the line \(3x + 4y = 5\), we can express this relationship as:
\[ 3h + 4k = 5 \]

### Step 2: Use the distance formula to find the radius
The radius \(r\) of the circle can be expressed using the distance from the center \((h, k)\) to the points \((1, -2)\) and \((4, -3)\). Therefore, we can write:
\[ r^2 = (h - 1)^2 + (k + 2)^2 \quad \text{(1)} \]
\[ r^2 = (h - 4)^2 + (k + 3)^2 \quad \text{(2)} \]

### Step 3: Set equations (1) and (2) equal to each other
Since both expressions equal \(r^2\), we can set them equal to each other:
\[ (h - 1)^2 + (k + 2)^2 = (h - 4)^2 + (k + 3)^2 \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ (h^2 - 2h + 1) + (k^2 + 4k + 4) = (h^2 - 8h + 16) + (k^2 + 6k + 9) \]

### Step 5: Simplify the equation
Cancelling \(h^2\) and \(k^2\) from both sides results in:
\[ -2h + 5 + 4k = -8h + 25 + 6k \]
Rearranging gives:
\[ 6h - 2k = 20 \]
Dividing through by 2 simplifies to:
\[ 3h - k = 10 \quad \text{(3)} \]

### Step 6: Solve the system of equations
Now we have a system of equations:
1. \(3h + 4k = 5\) (from the line)
2. \(3h - k = 10\) (from the distances)

We can solve these equations simultaneously. From equation (3), we can express \(k\) in terms of \(h\):
\[ k = 3h - 10 \]

Substituting this into equation (1):
\[ 3h + 4(3h - 10) = 5 \]
\[ 3h + 12h - 40 = 5 \]
\[ 15h = 45 \implies h = 3 \]

### Step 7: Find \(k\)
Now substituting \(h = 3\) back into the equation for \(k\):
\[ k = 3(3) - 10 = 9 - 10 = -1 \]

### Step 8: Find the radius
Now we have the center of the circle \((3, -1)\). We can find the radius by using either point. Using point \((1, -2)\):
\[ r^2 = (3 - 1)^2 + (-1 + 2)^2 = 2^2 + 1^2 = 4 + 1 = 5 \]

### 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 \(h = 3\), \(k = -1\), and \(r^2 = 5\):
\[ (x - 3)^2 + (y + 1)^2 = 5 \]

### Final Answer
The equation of the circle is:
\[ (x - 3)^2 + (y + 1)^2 = 5 \]