The co-ordinates of the end points of a diameter of a circle are (0,0) and (4,4). The equation of the circle is :

Options

A. `x^(2)+y^(2)-8x-8y=0`
B. `x^(2)+y^(2)+8x+8y=0`
C. `x^(2)+y^(2)+4x+4y=0`
D. `x^(2)+y^(2)-4x-4y=0`

Answer

To find the equation of the circle given the endpoints of its diameter, we can follow these steps:

### Step 1: Identify the coordinates of the endpoints of the diameter
The endpoints of the diameter are given as (0, 0) and (4, 4). We can denote these points as:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (4, 4) \)

### Step 2: Use the standard equation of the circle
The standard equation of a circle given the endpoints of the diameter is:
\[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \]
Substituting the values of \(x_1, y_1, x_2, y_2\):
\[ (x - 0)(x - 4) + (y - 0)(y - 4) = 0 \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ x(x - 4) + y(y - 4) = 0 \]
Expanding this gives:
\[ x^2 - 4x + y^2 - 4y = 0 \]

### Step 4: Rearranging the equation
We can rearrange the equation to a standard form:
\[ x^2 + y^2 - 4x - 4y = 0 \]

### Step 5: Final equation of the circle
Thus, the final equation of the circle is:
\[ x^2 + y^2 - 4x - 4y = 0 \]