Find the equation of the circle which passes through the points (1 ,-2 ), (4, - 3) and has its centre on the line 3x + 4y + 10 = 0.

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 given by the equation \(3x + 4y + 10 = 0\), we can follow these steps:

### Step 1: General Equation of a 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 (1, -2)
Substituting the point \((1, -2)\) into the circle's equation:
\[ 1^2 + (-2)^2 + 2g(1) + 2f(-2) + c = 0 \]
This simplifies to:
\[ 1 + 4 + 2g - 4f + c = 0 \]
or
\[ 2g - 4f + c + 5 = 0 \quad \text{(Equation 1)} \]

### Step 3: Substitute the Second Point (4, -3)
Now substituting the point \((4, -3)\):
\[ 4^2 + (-3)^2 + 2g(4) + 2f(-3) + c = 0 \]
This simplifies to:
\[ 16 + 9 + 8g - 6f + c = 0 \]
or
\[ 8g - 6f + c + 25 = 0 \quad \text{(Equation 2)} \]

### Step 4: Center on the Line
The center of the circle \((-g, -f)\) must satisfy the line equation \(3x + 4y + 10 = 0\). Substituting \(-g\) and \(-f\) into this equation gives:
\[ 3(-g) + 4(-f) + 10 = 0 \]
which simplifies to:
\[ -3g - 4f + 10 = 0 \quad \text{(Equation 3)} \]

### Step 5: Solve Equations
Now we have three equations:
1. \(2g - 4f + c + 5 = 0\) (Equation 1)
2. \(8g - 6f + c + 25 = 0\) (Equation 2)
3. \(-3g - 4f + 10 = 0\) (Equation 3)

### Step 6: Eliminate \(c\)
Subtract Equation 1 from Equation 2:
\[ (8g - 6f + c + 25) - (2g - 4f + c + 5) = 0 \]
This simplifies to:
\[ 6g - 2f + 20 = 0 \quad \Rightarrow \quad 3g - f + 10 = 0 \quad \text{(Equation 4)} \]

### Step 7: Solve Equations 3 and 4
Now we solve Equations 3 and 4 together:
1. \(-3g - 4f + 10 = 0\) (Equation 3)
2. \(3g - f + 10 = 0\) (Equation 4)

From Equation 4, we can express \(f\):
\[ f = 3g + 10 \]

Substituting this into Equation 3:
\[ -3g - 4(3g + 10) + 10 = 0 \]
This simplifies to:
\[ -3g - 12g - 40 + 10 = 0 \quad \Rightarrow \quad -15g - 30 = 0 \quad \Rightarrow \quad g = -2 \]

### Step 8: Find \(f\)
Substituting \(g = -2\) back into Equation 4:
\[ f = 3(-2) + 10 = -6 + 10 = 4 \]

### Step 9: Find \(c\)
Now substitute \(g\) and \(f\) back into Equation 1 to find \(c\):
\[ 2(-2) - 4(4) + c + 5 = 0 \]
This simplifies to:
\[ -4 - 16 + c + 5 = 0 \quad \Rightarrow \quad c - 15 = 0 \quad \Rightarrow \quad c = 15 \]

### Step 10: Write the Circle's Equation
Now we have \(g = -2\), \(f = 4\), and \(c = 15\). The equation of the circle is:
\[ x^2 + y^2 - 4x + 8y + 15 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 4x + 8y + 15 = 0 \]