The end points of a diameter of a circle are (1,-1) and (3,5). Find the equation of the circle. Also find its centre and radius.

Answer

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

### Step 1: Identify the endpoints of the diameter
The endpoints of the diameter are given as:
- Point A (x₁, y₁) = (1, -1)
- Point B (x₂, y₂) = (3, 5)

### Step 2: Calculate the center of the circle
The center (h, k) of the circle can be found by taking the midpoint of the diameter. The midpoint formula is given by:
\[ h = \frac{x₁ + x₂}{2}, \quad k = \frac{y₁ + y₂}{2} \]

Substituting the values:
\[ h = \frac{1 + 3}{2} = \frac{4}{2} = 2 \]
\[ k = \frac{-1 + 5}{2} = \frac{4}{2} = 2 \]

Thus, the center of the circle is (2, 2).

### Step 3: Calculate the radius of the circle
The radius (r) can be calculated as half the length of the diameter. First, we need to find the length of the diameter using the distance formula:
\[ d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²} \]

Substituting the values:
\[ d = \sqrt{(3 - 1)² + (5 - (-1))²} = \sqrt{(2)² + (6)²} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]

Since the radius is half of the diameter:
\[ r = \frac{d}{2} = \frac{2\sqrt{10}}{2} = \sqrt{10} \]

### Step 4: Write the equation of the circle
The standard equation of a circle with center (h, k) and radius r is:
\[ (x - h)² + (y - k)² = r² \]

Substituting the values of h, k, and r:
\[ (x - 2)² + (y - 2)² = (\sqrt{10})² \]
\[ (x - 2)² + (y - 2)² = 10 \]

### Final Result
The equation of the circle is:
\[ (x - 2)² + (y - 2)² = 10 \]
The center of the circle is (2, 2) and the radius is \(\sqrt{10}\).