Find the equation of the circle drawn on the line joining (-1,2) and (3, -4) as diameter.

Answer

To find the equation of the circle drawn on the line joining the points (-1, 2) and (3, -4) as the diameter, we can follow these steps:

### Step 1: Identify the endpoints of the diameter
The endpoints of the diameter are given as:
- Point 1 (x1, y1) = (-1, 2)
- Point 2 (x2, y2) = (3, -4)

### Step 2: Use the equation of the circle in diametric form
The equation of a circle in diametric form is given by:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

### Step 3: Substitute the values of the points into the equation
Substituting the coordinates of the points into the equation:
\[ (x + 1)(x - 3) + (y - 2)(y + 4) = 0 \]

### Step 4: Expand the equation
Now we will expand both products:
1. For \( (x + 1)(x - 3) \):
\[ x^2 - 3x + x - 3 = x^2 - 2x - 3 \]

2. For \( (y - 2)(y + 4) \):
\[ y^2 + 4y - 2y - 8 = y^2 + 2y - 8 \]

Putting it all together:
\[ x^2 - 2x - 3 + y^2 + 2y - 8 = 0 \]

### Step 5: Combine like terms
Combining the terms gives:
\[ x^2 + y^2 - 2x + 2y - 11 = 0 \]

### Step 6: Final equation of the circle
Thus, the equation of the circle is:
\[ x^2 + y^2 - 2x + 2y - 11 = 0 \]