A right angled isosceles triangle is inscribed in the circle `x^2 + y^2 _ 4x - 2y - 4 = 0` then length of its side is

Options

A. `3 sqrt(2)`
B. `2 sqrt(2)`
C. `sqrt(2)`
D. `4 sqrt(2)`

Explanation

To find the length of the side of a right-angled isosceles triangle inscribed in the given circle, we will follow these steps:

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 2y - 4 = 0 \]
We will rearrange this equation into the standard form by completing the square.

### Step 2: Completing the Square
1. For the \(x\) terms:
\[ x^2 - 4x \quad \text{(take half of -4, square it: } (-2)^2 = 4\text{)} \]
\[ = (x - 2)^2 - 4 \]

2. For the \(y\) terms:
\[ y^2 - 2y \quad \text{(take half of -2, square it: } (-1)^2 = 1\text{)} \]
\[ = (y - 1)^2 - 1 \]

Now substituting back into the equation:
\[ (x - 2)^2 - 4 + (y - 1)^2 - 1 - 4 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 1)^2 = 9 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\):
- Center \(C(2, 1)\)
- Radius \(r = \sqrt{9} = 3\)

### Step 4: Properties of the Inscribed Triangle
For a right-angled isosceles triangle inscribed in a circle, the hypotenuse is the diameter of the circle. Therefore, the length of the hypotenuse is:
\[ \text{Diameter} = 2 \times r = 2 \times 3 = 6 \]

### Step 5: Relate the Sides of the Triangle
Let the length of each equal side of the isosceles triangle be \(x\). According to the Pythagorean theorem:
\[ x^2 + x^2 = 6^2 \]
\[ 2x^2 = 36 \]
\[ x^2 = 18 \]
\[ x = \sqrt{18} = 3\sqrt{2} \]

### Conclusion
The length of each side of the right-angled isosceles triangle is:
\[ \boxed{3\sqrt{2}} \]