The equation of a circle whose end points of a diameter are `(x_(1) , y_(1))` and `(x_(2) y_(2))` is

Options

A. `(x - x_(1)) (x - x_(2)) + (y - y_(1)) (y - y_(2)) = x^(2) + y^(2)`
B. `(x - x_(1))^(2) + (y- y_(1))^(2) = x_(2) + y_(2)`
C. `x^(2) + y^(2) + 2x_(1) x _(2) + 2y_(1) y_(2) = 0`
D. `(x - x_(1)) (x- x_(2)) + (y - y_(1)) (y - y_(2)) =0`

Answer

To find the equation of a circle whose endpoints of a diameter are given as \((x_1, y_1)\) and \((x_2, y_2)\), we can follow these steps:

### Step 1: Understand the Formula
The general equation of a circle with endpoints of a diameter at \((x_1, y_1)\) and \((x_2, y_2)\) can be derived from the fact that any point \((x, y)\) on the circle satisfies the equation:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

### Step 2: Write the Equation
Using the formula from Step 1, we can directly write the equation of the circle:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]

### Step 3: Expand the Equation
To express the equation in a standard form, we can expand the equation:
1. Expand \((x - x_1)(x - x_2)\):
\[ x^2 - (x_1 + x_2)x + x_1x_2 \]
2. Expand \((y - y_1)(y - y_2)\):
\[ y^2 - (y_1 + y_2)y + y_1y_2 \]
3. Combine the two expansions:
\[ x^2 - (x_1 + x_2)x + x_1x_2 + y^2 - (y_1 + y_2)y + y_1y_2 = 0 \]

### Step 4: Rearrange the Equation
Rearranging gives us the standard form of the circle's equation:
\[ x^2 + y^2 - (x_1 + x_2)x - (y_1 + y_2)y + (x_1x_2 + y_1y_2) = 0 \]

### Final Equation
Thus, the equation of the circle whose endpoints of a diameter are \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]