STATEMENT-1 : Tangent at any point `P(x_(1), y_(1))` on the hyperbola` xy = c^(2)` meets the co-ordinate axes at points Q and R, the circumcentre of `triangleOQR` has co-ordinate `(x_(1)y_(1))` .
and
STATEMENT-2 : Equation of tangent at point `(x_(1)y_(1))` to the curve `xy = c^(2)` is `(x)/(x_(1)) + (y)/(y_(1)) =2`.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a 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 true

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 2
**Statement 2:** The equation of the tangent at point \( P(x_1, y_1) \) to the curve \( xy = c^2 \) is given by \( \frac{x}{x_1} + \frac{y}{y_1} = 2 \).

1. **Find the slope of the tangent:**
- The equation of the hyperbola is \( xy = c^2 \).
- Differentiate implicitly:
\[ x \frac{dy}{dx} + y = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
- At point \( P(x_1, y_1) \):
\[ \text{slope} = -\frac{y_1}{x_1} \]

2. **Use point-slope form to find the equation of the tangent:**
- The point-slope form is:
\[ y - y_1 = m(x - x_1) \]
- Substituting the slope:
\[ y - y_1 = -\frac{y_1}{x_1}(x - x_1) \]
- Rearranging gives:
\[ y - y_1 = -\frac{y_1}{x_1}x + y_1 \]
- Simplifying:
\[ y = -\frac{y_1}{x_1}x + 2y_1 \]
- Multiplying through by \( x_1y_1 \):
\[ yx_1 + y_1x = 2x_1y_1 \]
- Rearranging gives:
\[ \frac{x}{x_1} + \frac{y}{y_1} = 2 \]
- Thus, **Statement 2 is correct**.

### Step 2: Analyze Statement 1
**Statement 1:** The circumcenter of triangle \( OQR \) has coordinates \( (x_1, y_1) \).

1. **Find the intercepts of the tangent:**
- Set \( y = 0 \) to find the x-intercept:
\[ 0 = -\frac{y_1}{x_1}x + 2y_1 \implies x = 2x_1 \]
- Set \( x = 0 \) to find the y-intercept:
\[ y = 2y_1 \]
- Thus, the coordinates of points \( Q \) and \( R \) are \( Q(2x_1, 0) \) and \( R(0, 2y_1) \).

2. **Find the circumcenter of triangle \( OQR \):**
- The coordinates of \( O \) are \( (0, 0) \), \( Q(2x_1, 0) \), and \( R(0, 2y_1) \).
- The circumcenter of triangle formed by points \( O(0,0) \), \( Q(2x_1, 0) \), and \( R(0, 2y_1) \) can be found as the midpoint of the hypotenuse \( QR \).
- The midpoint of \( QR \):
\[ \left( \frac{2x_1 + 0}{2}, \frac{0 + 2y_1}{2} \right) = (x_1, y_1) \]
- Therefore, **Statement 1 is also correct**.

### Conclusion
Both statements are correct, but Statement 2 is not a correct explanation of Statement 1. Therefore, the final answer is:

**Statement 1 is true, Statement 2 is true, but Statement 2 is not the correct explanation of Statement 1.**