The co-ordinates of the end points of a diameter of a circle are (3, -2) and (-3, 6). Find the co-ordinates of the centre and radius.

Answer

To find the coordinates of the center and the radius of the circle given the endpoints of its diameter, we can follow these steps:

### Step 1: Find the Coordinates of the Center
The center of the circle can be found by calculating the midpoint of the diameter. The midpoint \( M(x, y) \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Given the endpoints of the diameter are \( (3, -2) \) and \( (-3, 6) \):

- \( x_1 = 3, y_1 = -2 \)
- \( x_2 = -3, y_2 = 6 \)

Now, substituting these values into the midpoint formula:

\[ M = \left( \frac{3 + (-3)}{2}, \frac{-2 + 6}{2} \right) \]

Calculating the x-coordinate:

\[ M_x = \frac{3 - 3}{2} = \frac{0}{2} = 0 \]

Calculating the y-coordinate:

\[ M_y = \frac{-2 + 6}{2} = \frac{4}{2} = 2 \]

Thus, the coordinates of the center are \( (0, 2) \).

### Step 2: Find the Radius
The radius of the circle is half the length of the diameter. To find the length of the diameter, we can use the distance formula between the two endpoints:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the endpoints:

\[ d = \sqrt{((-3) - 3)^2 + (6 - (-2))^2} \]

Calculating the differences:

\[ d = \sqrt{(-6)^2 + (8)^2} \]

Calculating the squares:

\[ d = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Now, the radius \( r \) is half of the diameter:

\[ r = \frac{d}{2} = \frac{10}{2} = 5 \]

### Final Answer
- The coordinates of the center of the circle are \( (0, 2) \).
- The radius of the circle is \( 5 \).

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