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 solve the problem, we need to find the coordinates of the center and the radius of the circle given the endpoints of its diameter. The endpoints are (3, -2) and (-3, 6).

### Step-by-Step Solution:

**Step 1: Find the coordinates of the center of the circle.**

The center of the circle can be found using the midpoint formula. The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

In our case, the endpoints are \( (3, -2) \) and \( (-3, 6) \).

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

Now, substituting these values into the midpoint formula:

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

Calculating the x-coordinate:

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

Calculating the y-coordinate:

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

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

**Step 2: Find the radius of the circle.**

The radius is the distance from the center to one of the endpoints. We can use the distance formula, which is given by:

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

We will find the distance from the center \( C(0, 2) \) to one of the endpoints, say \( A(3, -2) \).

Let:
- \( (x_1, y_1) = (0, 2) \)
- \( (x_2, y_2) = (3, -2) \)

Now, substituting these values into the distance formula:

\[ d = \sqrt{(3 - 0)^2 + (-2 - 2)^2} \]

Calculating the x-component:

\[ (3 - 0)^2 = 3^2 = 9 \]

Calculating the y-component:

\[ (-2 - 2)^2 = (-4)^2 = 16 \]

Now, substituting back into the distance formula:

\[ d = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Thus, the radius of the circle is \( 5 \) units.

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

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