Find the length of latus rectum L of ellipse `(x^(2))/(A^(2)) + (y^(2))/(9) = 1, (A^2) gt 9)` where`A^2` is the area enclosed by quadrilateral formed by joining the focii of hyperbola `(x^(2))/(16) - (y^(2))/(9) = 1` and its conjugate hyperbola

Options

A. `(18sqrt(2))/(5)`
B. `(5)/(18)sqrt(2)`
C. `(18)/(5sqrt(2))`
D. `(9)/(5sqrt(2))`

Explanation

To solve the problem step-by-step, we need to find the length of the latus rectum \( L \) of the ellipse given by the equation

\[ \frac{x^2}{A^2} + \frac{y^2}{9} = 1 \]

where \( A^2 \) is the area enclosed by the quadrilateral formed by joining the foci of the hyperbola

\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]

and its conjugate hyperbola.

### Step 1: Identify the foci of the hyperbola
For the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \):
- The value of \( a^2 = 16 \) and \( b^2 = 9 \).
- The foci are located at \( (\pm ae, 0) \), where \( e \) is the eccentricity.

### Step 2: Calculate the eccentricity \( e \)
The eccentricity \( e \) for a hyperbola is given by:

\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \]

### Step 3: Calculate \( a \cdot e \)
Now, we calculate \( a \cdot e \):

\[ a = \sqrt{16} = 4 \]
\[ a \cdot e = 4 \cdot \frac{5}{4} = 5 \]

### Step 4: Calculate the area of the quadrilateral
The area \( A \) of the quadrilateral formed by the foci and the intersections with the conjugate hyperbola can be calculated as follows:

The distance between the foci is \( 2ae \):

\[ 2ae = 2 \cdot 4 \cdot \frac{5}{4} = 10 \]

The area of the quadrilateral formed by the foci is given by:

\[ A = 2 \cdot (a \cdot e) = 2 \cdot 5 = 10 \]

### Step 5: Find \( A^2 \)
Since the area \( A \) is \( 10 \), we have:

\[ A^2 = 10^2 = 100 \]

### Step 6: Find the value of \( A \) for the ellipse
Now, we need to find the length of the latus rectum \( L \) of the ellipse given by:

\[ \frac{x^2}{A^2} + \frac{y^2}{9} = 1 \]

### Step 7: Length of the latus rectum
The length of the latus rectum \( L \) of an ellipse is given by:

\[ L = \frac{2b^2}{a} \]

In our case, \( b^2 = 9 \) and \( A^2 = 100 \), so \( A = 10 \).

### Step 8: Substitute values into the formula
Now substituting the values into the formula:

\[ L = \frac{2 \cdot 9}{10} = \frac{18}{10} = \frac{9}{5} \]

### Final Answer
Thus, the length of the latus rectum \( L \) is:

\[ L = \frac{9}{5} \]