Statement:1 There is only one circle which touches the curve `xy =1 at (1,1)` and x-axis. Because
Statement 2: Any circle touching the hypebola `xy=1 at (1,1) ` can be written in the from `(x-1)^(2) + (y-1) ^(2) + lamda( y+x-2) =0.`

Options

A. Stateme-1 is True, Statement-2 is True, Statemetn-2 is correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is ture

Answer

To solve the problem, we need to analyze the two statements provided and derive the necessary equations step by step.

### Step 1: Understand the Problem
We have two statements:
1. There is only one circle which touches the curve \(xy = 1\) at the point \((1, 1)\) and the x-axis.
2. Any circle touching the hyperbola \(xy = 1\) at \((1, 1)\) can be written in the form \((x-1)^2 + (y-1)^2 + \lambda(y+x-2) = 0\).

### Step 2: Find the Equation of the Tangent to the Hyperbola
The hyperbola is given by \(xy = 1\). To find the tangent at the point \((1, 1)\):
- The slope of the hyperbola at \((1, 1)\) can be found using implicit differentiation:
\[ \frac{dy}{dx} = -\frac{y}{x^2} \]
At \((1, 1)\), this gives:
\[ \frac{dy}{dx} = -1 \]
- The equation of the tangent line at \((1, 1)\) is:
\[ y - 1 = -1(x - 1) \implies y + x - 2 = 0 \]

### Step 3: Write the General Equation of the Circle
The general equation of a circle that touches the tangent line can be expressed as:
\[ (x - 1)^2 + (y - 1)^2 + \lambda(y + x - 2) = 0 \]
Expanding this, we get:
\[ x^2 - 2x + 1 + y^2 - 2y + 1 + \lambda(y + x - 2) = 0 \]
This simplifies to:
\[ x^2 + y^2 + (\lambda - 2)x + (\lambda - 2)y + 2 = 0 \]

### Step 4: Determine Conditions for the Circle to Touch the x-axis
For the circle to touch the x-axis, the distance from the center of the circle to the x-axis must equal the radius. The center of the circle is at \((1, 1)\) with respect to the equation we derived.

The radius \(r\) can be expressed as:
\[ r = \sqrt{g^2 + f^2 - c} \]
where \(g = \frac{\lambda - 2}{2}\), \(f = \frac{\lambda - 2}{2}\), and \(c = -2\).

### Step 5: Set Up the Equation for Touching the x-axis
The condition for touching the x-axis is:
\[ \text{y-coordinate of center} = \text{radius} \]
Thus:
\[ \frac{\lambda - 2}{2} = \sqrt{\left(\frac{\lambda - 2}{2}\right)^2 + \left(\frac{\lambda - 2}{2}\right)^2 + 2} \]

### Step 6: Solve for \(\lambda\)
Squaring both sides and simplifying will yield a quadratic equation in \(\lambda\):
\[ \lambda^2 - 4\lambda + 4 = 0 \]
This has two solutions, indicating that there are two circles that can touch the hyperbola at the point \((1, 1)\) and the x-axis.

### Conclusion
- **Statement 1** is incorrect because there are two circles, not one.
- **Statement 2** is correct as it correctly describes the form of the circles.

### Final Answer
- Statement 1: False
- Statement 2: True