Area of the quadrilateral formed with the foci of the hyperbola `x^2/a^2-y^2/b^2=1 and x^2/a^2-y^2/b^2=-1` (a) `4(a^2+b^2)` (b) `2(a^2+b^2)` (c) `(a^2+b^2)` (d) `1/2(a^2+b^2)`

Options

A. `4(a^(2)+b^(2))`
B. `2(a^(2)+b^(2))`
C. `(a^(2)+b^(2)`
D. `(1)/(2)(a^(2)+b^(2))`

Answer

To find the area of the quadrilateral formed by the foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and its conjugate hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \), we can follow these steps:

### Step 1: Determine the foci of the hyperbola
The foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by the coordinates \( (ae, 0) \) and \( (-ae, 0) \), where \( e = \sqrt{1 + \frac{b^2}{a^2}} \).

- **Hint**: Remember that the foci are located along the x-axis for this hyperbola.

### Step 2: Determine the foci of the conjugate hyperbola
The foci of the conjugate hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \) are given by the coordinates \( (0, be) \) and \( (0, -be) \), where \( e = \sqrt{1 + \frac{a^2}{b^2}} \).

- **Hint**: The foci of the conjugate hyperbola will be located along the y-axis.

### Step 3: Identify the coordinates of the foci
From the above steps, we have the following coordinates for the foci:
- For the hyperbola: \( F_1(ae, 0) \) and \( F_2(-ae, 0) \)
- For the conjugate hyperbola: \( F_3(0, be) \) and \( F_4(0, -be) \)

- **Hint**: Make sure to label the foci clearly to avoid confusion later.

### Step 4: Calculate the area of the quadrilateral formed by the foci
The quadrilateral formed by these four points \( F_1, F_2, F_3, F_4 \) is a rhombus. The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

### Step 5: Find the lengths of the diagonals
- The length of diagonal \( d_1 \) (between \( F_1 \) and \( F_2 \)) is \( 2ae \).
- The length of diagonal \( d_2 \) (between \( F_3 \) and \( F_4 \)) is \( 2be \).

### Step 6: Substitute the lengths into the area formula
Substituting \( d_1 \) and \( d_2 \) into the area formula:
\[ A = \frac{1}{2} \times (2ae) \times (2be) = 2ab \cdot e^2 \]

### Step 7: Substitute the value of \( e^2 \)
We know:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
Thus:
\[ A = 2ab \cdot \left(1 + \frac{b^2}{a^2}\right) = 2ab + \frac{2b^3}{a} \]

### Step 8: Final area expression
After simplifying, we can express the area in terms of \( a^2 \) and \( b^2 \):
\[ A = 2(a^2 + b^2) \]

### Conclusion
Thus, the area of the quadrilateral formed by the foci of the hyperbola and its conjugate hyperbola is:
\[ \boxed{2(a^2 + b^2)} \]