The equation of a circle whose diameter is the line joining the points `(-4,3) and (12, -1)` is

Options

A. `x ^(2) + y^(2) + 8x + 2y + 51=0`
B. `x ^(2) + y^(2)+ 8x -2y -51=0`
C. `x ^(2) + y^(2) + 8x + 2y -51=0`
D. `x ^(2) + y ^(2) -8x - 2y -51 =0`

Answer

To find the equation of a circle whose diameter is defined by the endpoints \((-4, 3)\) and \((12, -1)\), we can follow these steps:

### Step 1: Identify the endpoints of the diameter
Let the points be:
- \( A(-4, 3) \) which is \((x_1, y_1)\)
- \( B(12, -1) \) which is \((x_2, y_2)\)

### Step 2: Write the general equation of the circle using the diameter endpoints
The equation of a circle based on the endpoints of the diameter is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

### Step 3: Substitute the values of \(x_1, y_1, x_2, y_2\)
Substituting the coordinates of points \(A\) and \(B\):
\[ (x - (-4))(x - 12) + (y - 3)(y - (-1)) = 0 \]
This simplifies to:
\[ (x + 4)(x - 12) + (y - 3)(y + 1) = 0 \]

### Step 4: Expand the equation
Now we will expand both products:
1. Expanding \((x + 4)(x - 12)\):
\[ x^2 - 12x + 4x - 48 = x^2 - 8x - 48 \]

2. Expanding \((y - 3)(y + 1)\):
\[ y^2 + y - 3y - 3 = y^2 - 2y - 3 \]

### Step 5: Combine the expanded parts
Now, combine both expanded results:
\[ (x^2 - 8x - 48) + (y^2 - 2y - 3) = 0 \]
This simplifies to:
\[ x^2 + y^2 - 8x - 2y - 51 = 0 \]

### Step 6: Final equation of the circle
Thus, the equation of the circle is:
\[ x^2 + y^2 - 8x - 2y - 51 = 0 \]