Find the equation of a circle passing through the points (-1,5) and (-7,5) and its centre lies on the line 3x+4y+8=0.

Answer

To find the equation of a circle passing through the points (-1, 5) and (-7, 5) with its center lying on the line \(3x + 4y + 8 = 0\), we can follow these steps:

### Step 1: Identify the coordinates of the points
The points through which the circle passes are:
- Point A: \((-1, 5)\)
- Point B: \((-7, 5)\)

### Step 2: Determine the center of the circle
Let the center of the circle be \(C(a, b)\). Since the center lies on the line \(3x + 4y + 8 = 0\), we can substitute \(a\) and \(b\) into this equation:
\[ 3a + 4b + 8 = 0 \quad \text{(Equation 1)} \]

### Step 3: Use the property of equal distances
The distances from the center \(C\) to points A and B must be equal:
\[ AC = BC \]
Calculating the distances:
\[ AC = \sqrt{(a + 1)^2 + (b - 5)^2} \]
\[ BC = \sqrt{(a + 7)^2 + (b - 5)^2} \]
Setting these equal gives:
\[ \sqrt{(a + 1)^2 + (b - 5)^2} = \sqrt{(a + 7)^2 + (b - 5)^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ (a + 1)^2 + (b - 5)^2 = (a + 7)^2 + (b - 5)^2 \]
The \((b - 5)^2\) terms cancel out:
\[ (a + 1)^2 = (a + 7)^2 \]

### Step 5: Expand both sides
Expanding gives:
\[ a^2 + 2a + 1 = a^2 + 14a + 49 \]
Cancelling \(a^2\) from both sides:
\[ 2a + 1 = 14a + 49 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 2a - 14a = 49 - 1 \]
\[ -12a = 48 \]
\[ a = -4 \]

### Step 7: Substitute \(a\) back into Equation 1
Substituting \(a = -4\) into Equation 1:
\[ 3(-4) + 4b + 8 = 0 \]
\[ -12 + 4b + 8 = 0 \]
\[ 4b - 4 = 0 \]
\[ 4b = 4 \]
\[ b = 1 \]

### Step 8: Determine the center of the circle
The center of the circle is \(C(-4, 1)\).

### Step 9: Calculate the radius
The radius \(r\) is the distance from the center \(C(-4, 1)\) to either point A or B. Using point A:
\[ r = AC = \sqrt{(-4 + 1)^2 + (1 - 5)^2} \]
\[ = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 10: Write the equation of the circle
The equation of the circle with center \((-4, 1)\) and radius \(5\) is:
\[ (x + 4)^2 + (y - 1)^2 = 25 \]

### Final Answer
The equation of the circle is:
\[ (x + 4)^2 + (y - 1)^2 = 25 \]