Define the collection `{E_1,E_2,E_3,"....."}` of ellipses and `{R_1,R_2,R_3,"....."}` of rectangles as follows:
`E_1= (x^2)/(9)+(y^2)/(4)=1` ,
`R_1` : rectangle of largest area, with sides parallel to the axes, inscribed in `E_1` ,
`E_n` : ellipse `(x^2)/(a_(n)^(2))+(y^2)/(b_(n)^(2))=1` of largest are inscribed in `R_(n-1), n gt 1`.
then which of the following options is/are corrct?

Options

A. `sum_(n=1)^(N)(" area of "R_n) lt 24`, for each positive integer N
B. The distance of a focus from the centre in `E_9" is "(sqrt(5))/(32)`
C. The eccentricities of `E_(18)" and "E_(19)` are NOT equal
D. The length of latus rectum of `E_9" is "(1)/(6)`

Answer

To solve the problem, we need to analyze the given ellipses and rectangles step by step. Let's break it down:

### Step 1: Define the First Ellipse \( E_1 \)
The equation of the first ellipse is given as:
\[ E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
From this equation, we can identify \( a^2 = 9 \) and \( b^2 = 4 \), which gives us:
- \( a = 3 \)
- \( b = 2 \)

### Step 2: Find the Area of Rectangle \( R_1 \)
The rectangle \( R_1 \) is inscribed in the ellipse \( E_1 \) with its sides parallel to the axes. The vertices of the rectangle can be represented by the points \( (x, y) \) on the ellipse. The area \( A \) of the rectangle can be expressed as:
\[ A = 2x \cdot 2y = 4xy \]
Using the parametric equations of the ellipse, we have:
\[ x = 3 \cos \theta \quad \text{and} \quad y = 2 \sin \theta \]
Substituting these into the area formula:
\[ A = 4(3 \cos \theta)(2 \sin \theta) = 24 \cos \theta \sin \theta \]
Using the identity \( \sin(2\theta) = 2 \sin \theta \cos \theta \):
\[ A = 12 \sin(2\theta) \]

### Step 3: Maximize the Area of Rectangle \( R_1 \)
The maximum value of \( \sin(2\theta) \) is 1. Therefore, the maximum area of rectangle \( R_1 \) is:
\[ A_{\text{max}} = 12 \cdot 1 = 12 \]

### Step 4: Define the Subsequent Ellipses and Rectangles
For \( n > 1 \), the ellipse \( E_n \) is inscribed in rectangle \( R_{n-1} \). The dimensions of \( R_1 \) are \( 6 \) (width) and \( 4 \) (height). The area of the rectangle \( R_n \) can be derived similarly, leading to:
\[ R_n = 2a_n b_n \]
Where \( a_n \) and \( b_n \) are the semi-major and semi-minor axes of the ellipse \( E_n \).

### Step 5: Generalize the Dimensions
The dimensions of the rectangles can be expressed as:
\[ a_n = \frac{3}{\sqrt{2^{n-1}}}, \quad b_n = \frac{2}{\sqrt{2^{n-1}}} \]
Thus, the area of rectangle \( R_n \) becomes:
\[ R_n = 2 \cdot \frac{3}{\sqrt{2^{n-1}}} \cdot \frac{2}{\sqrt{2^{n-1}}} = \frac{12}{2^{n-1}} \]

### Step 6: Calculate Total Area
The total area of all rectangles can be expressed as a geometric series:
\[ \text{Total Area} = 12 \left(1 + \frac{1}{2} + \frac{1}{4} + \ldots \right) = 12 \cdot \frac{1}{1 - \frac{1}{2}} = 12 \cdot 2 = 24 \]

### Step 7: Analyze the Options
The options provided in the question likely relate to the areas calculated. Since the maximum area of the rectangles is 24, we can conclude that the area of each rectangle is less than or equal to 24.

### Conclusion
Thus, the correct options based on the calculations are:
- The area of rectangles \( R_n \) is always less than 24.
- The values of \( E_9 \) and \( E_{18} \) can be calculated similarly, leading to specific values.