A right angled isoceles triangle is inscribed in the circle `x^(2)+y^(2)-6x+10y-38=0` then its area is (square units)

Options

A. 18
B. 12
C. 72
D. 36

Answer

To solve the problem of finding the area of a right-angled isosceles triangle inscribed in the given circle, we will follow these steps:

### Step 1: Write the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 6x + 10y - 38 = 0 \]

We can rearrange this equation to find the center and radius of the circle. We will complete the square for both \(x\) and \(y\).

### Step 2: Complete the square
1. For \(x\):
\[ x^2 - 6x \quad \text{can be rewritten as} \quad (x-3)^2 - 9 \]

2. For \(y\):
\[ y^2 + 10y \quad \text{can be rewritten as} \quad (y+5)^2 - 25 \]

Now substituting these back into the equation:
\[ (x-3)^2 - 9 + (y+5)^2 - 25 - 38 = 0 \]
This simplifies to:
\[ (x-3)^2 + (y+5)^2 - 72 = 0 \]
Thus, we can rewrite it as:
\[ (x-3)^2 + (y+5)^2 = 72 \]

### Step 3: Identify the center and radius of the circle
From the standard form \((x-h)^2 + (y-k)^2 = r^2\), we can identify:
- Center \((h, k) = (3, -5)\)
- Radius \(r = \sqrt{72} = 6\sqrt{2}\)

### Step 4: Determine the side length of the triangle
Since the triangle is a right-angled isosceles triangle, the hypotenuse will be the diameter of the circle. The diameter \(d\) is twice the radius:
\[ d = 2r = 2 \times 6\sqrt{2} = 12\sqrt{2} \]

Let the equal sides of the triangle be \(a\). According to the Pythagorean theorem for a right-angled triangle:
\[ d^2 = a^2 + a^2 = 2a^2 \]
Thus, we have:
\[ (12\sqrt{2})^2 = 2a^2 \]
Calculating the left side:
\[ 144 \times 2 = 288 \]
So, we have:
\[ 288 = 2a^2 \implies a^2 = 144 \implies a = 12 \]

### Step 5: Calculate the area of the triangle
The area \(A\) of an isosceles right triangle is given by:
\[ A = \frac{1}{2} \times a \times a = \frac{1}{2} a^2 \]
Substituting \(a = 12\):
\[ A = \frac{1}{2} \times 12^2 = \frac{1}{2} \times 144 = 72 \]

### Conclusion
The area of the inscribed right-angled isosceles triangle is:
\[ \boxed{72} \text{ square units} \]