Tangents at P to rectangular hyperbola `xy=2` meets coordinate axes at A and B, then area of triangle OAB (where O is origin) is _____.

Answer

To find the area of triangle OAB formed by the tangents from point P on the rectangular hyperbola \( xy = 2 \) to the coordinate axes, we can follow these steps:

### Step 1: Identify the coordinates of point P
Let the coordinates of point P be \( (x_1, y_1) \). Since P lies on the hyperbola, it satisfies the equation:
\[ x_1 y_1 = 2 \]

### Step 2: Find the slope of the tangent at point P
The slope of the tangent to the hyperbola at point \( (x_1, y_1) \) can be found using implicit differentiation. The equation of the hyperbola is:
\[ xy = 2 \]
Differentiating both sides with respect to \( x \) using the product rule gives:
\[ y + x \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
At point \( P \), the slope \( m \) is:
\[ m = -\frac{y_1}{x_1} \]

### Step 3: Write the equation of the tangent line at point P
The equation of the tangent line at point \( P \) can be expressed in point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting the slope:
\[ y - y_1 = -\frac{y_1}{x_1}(x - x_1) \]
Rearranging gives:
\[ y_1 x - x_1 y + x_1 y_1 = 0 \]

### Step 4: Find the coordinates of points A and B
**Point A** (where the tangent meets the x-axis, \( y = 0 \)):
Substituting \( y = 0 \) into the tangent equation:
\[ y_1 x - x_1 (0) + x_1 y_1 = 0 \implies y_1 x + x_1 y_1 = 0 \implies x = 2x_1 \]
Thus, the coordinates of point A are \( (2x_1, 0) \).

**Point B** (where the tangent meets the y-axis, \( x = 0 \)):
Substituting \( x = 0 \) into the tangent equation:
\[ y_1 (0) - x_1 y + x_1 y_1 = 0 \implies -x_1 y + x_1 y_1 = 0 \implies y = 2y_1 \]
Thus, the coordinates of point B are \( (0, 2y_1) \).

### Step 5: Calculate the area of triangle OAB
The area \( A \) of triangle OAB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \( OA = 2x_1 \) and the height \( OB = 2y_1 \):
\[ \text{Area} = \frac{1}{2} \times (2x_1) \times (2y_1) = 2x_1y_1 \]

### Step 6: Substitute the value of \( x_1y_1 \)
Since \( x_1y_1 = 2 \) (from the hyperbola equation):
\[ \text{Area} = 2 \times 2 = 4 \]

### Conclusion
The area of triangle OAB is:
\[ \boxed{4} \text{ square units} \]