Find the whole area included between the curve `x^2)y^(2) = a^(2)(y^(2) – x^(2)) `and its asymptotes (asymptotes are the lines which meet the curve at infinity).

Answer

To find the area included between the curve \(x^2 y^2 = a^2 (y^2 - x^2)\) and its asymptotes, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^2 y^2 = a^2 (y^2 - x^2) \]
Rearranging this, we get:
\[ x^2 y^2 + a^2 x^2 - a^2 y^2 = 0 \]
This can be rewritten as:
\[ a^2 y^2 - (1 + a^2) x^2 = 0 \]

### Step 2: Identify the asymptotes
To find the asymptotes, we set the equation to zero:
\[ a^2 y^2 - (1 + a^2) x^2 = 0 \]
This implies:
\[ a^2 y^2 = (1 + a^2) x^2 \]
Dividing both sides by \(x^2\) (assuming \(x \neq 0\)):
\[ \frac{y^2}{x^2} = \frac{1 + a^2}{a^2} \]
Taking the square root gives:
\[ \frac{y}{x} = \pm \sqrt{\frac{1 + a^2}{a^2}} \Rightarrow y = \pm \frac{\sqrt{1 + a^2}}{a} x \]
Thus, the asymptotes are:
\[ y = \frac{\sqrt{1 + a^2}}{a} x \quad \text{and} \quad y = -\frac{\sqrt{1 + a^2}}{a} x \]

### Step 3: Determine the points of intersection
To find the points where the asymptotes intersect the curve, we can substitute \(y = kx\) (where \(k = \pm \frac{\sqrt{1 + a^2}}{a}\)) back into the original equation. However, since we are looking for the area between the curve and the asymptotes, we can directly analyze the area formed by these lines.

### Step 4: Sketch the region
The asymptotes divide the plane into four quadrants. The area enclosed by the asymptotes and the curve can be visualized as a square. The vertices of this square will be at the points where the asymptotes intersect the lines \(x = a\) and \(y = a\).

### Step 5: Calculate the area
The side length of the square formed by the intersection of the asymptotes can be found by determining the distance from the origin to the points where the asymptotes intersect the lines \(x = a\) and \(y = a\). The side length is given by:
\[ \text{Side length} = 2a \]
Thus, the area \(A\) of the square is:
\[ A = (\text{Side length})^2 = (2a)^2 = 4a^2 \]

### Final Answer
The area included between the curve and its asymptotes is:
\[ \boxed{4a^2} \]