Consider a hyperbola `xy = 4` and a line `y = 2x = 4`. O is the centre of hyperbola. Tangent at any point P of hyperbola intersect the coordinate axes at A and B.
Locus of circumcentre of triangle OAB is

Options

A. an ellipse with eccentricity `(1)/(sqrt(2))`
B. a parabola
C. a hyperbola with eccnetricity `sqrt(2)`
D. a circle

Explanation

To solve the problem, we need to find the locus of the circumcenter of triangle OAB formed by the origin O and the points A and B where the tangent to the hyperbola \(xy = 4\) intersects the coordinate axes.

### Step-by-Step Solution:

1. **Identify the Hyperbola and its Center**:
The hyperbola is given by the equation \(xy = 4\). The center O of the hyperbola is at the origin (0, 0).

2. **Find a General Point on the Hyperbola**:
A general point \(P\) on the hyperbola can be expressed in terms of a parameter \(t\):
\[ P = (2t, \frac{4}{t}) \]

3. **Equation of the Tangent at Point P**:
The equation of the tangent to the hyperbola \(xy = 4\) at the point \(P(2t, \frac{4}{t})\) is given by:
\[ x + \frac{4}{t}y = 4 \]

4. **Find Points A and B**:
- **Point A**: To find the x-intercept (where \(y = 0\)):
\[ x + 0 = 4 \implies x = 4 \quad \Rightarrow A(4, 0) \]
- **Point B**: To find the y-intercept (where \(x = 0\)):
\[ 0 + \frac{4}{t}y = 4 \implies y = t \quad \Rightarrow B(0, t) \]

5. **Coordinates of Points A and B**:
Thus, the coordinates of points A and B are:
\[ A(4, 0) \quad \text{and} \quad B(0, t) \]

6. **Find the Circumcenter of Triangle OAB**:
The circumcenter of triangle OAB (where O is the origin) can be found as the midpoint of segment AB:
\[ \text{Midpoint of AB} = \left(\frac{4 + 0}{2}, \frac{0 + t}{2}\right) = \left(2, \frac{t}{2}\right) \]

7. **Express t in terms of x and y**:
Since the coordinates of the circumcenter are \((2, \frac{t}{2})\), we can express \(t\) in terms of the y-coordinate:
\[ y = \frac{t}{2} \implies t = 2y \]

8. **Substitute t into the Hyperbola Equation**:
Substitute \(t = 2y\) into the hyperbola equation \(xy = 4\):
\[ x(2y) = 4 \implies 2xy = 4 \implies xy = 2 \]

9. **Locus of the Circumcenter**:
Thus, the locus of the circumcenter of triangle OAB is given by:
\[ xy = 2 \]

### Final Answer:
The locus of the circumcenter of triangle OAB is the equation of the hyperbola:
\[ xy = 2 \]