Define the collections {E₁, E₂, E₃, ...} of ellipses and {R₁, R₂, R₃, ...} of rectangles as follows:

Q: Wait, why is the eccentricity constant? Doesn't shrinking the ellipse change its shape?

That is a brilliant intuition, but let's look closer at the definition of eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$. Eccentricity depends entirely on the ratio of the semi-minor axis to the semi-major axis. Because both $a_n$ and $b_n$ are scaled down by the exact same factor of $\frac{1}{\sqrt{2}}$ at every step, the ratio $\frac{b_n}{a_n}$ remains invariant. The ellipse shrinks, but it remains geometrically similar to the original. It is a perfect, self-similar scaling!

Q: How do we know the axes of $E_{n+1}$ align with the axes of $R_n$?

For an ellipse to be inscribed in a rectangle with maximum area, it must touch the midpoints of the rectangle's sides. By symmetry, the axes of the ellipse must coincide with the axes of the rectangle. If they were tilted, the area would not be maximized, and the symmetry of the inscribed rectangle would be broken.

Q: Why is the sum of areas strictly less than 24?

The sum of the areas forms a geometric series: $12 + 6 + 3 + \dots$. This is a convergent series. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$. Here, $a = 12$ and $r = 0.5$. So, $S = \frac{12}{1 - 0.5} = 24$. Since we are summing a finite number of terms $N$, the total area will always be strictly less than the infinite limit of 24.

Question

Define the collections $\{E_1, E_2, E_3, \dots\}$ of ellipses and $\{R_1, R_2, R_3, \dots\}$ of rectangles as follows:
$E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1$;
$R_1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;
$E_n$ : ellipse $\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}, n > 1$;
$R_n$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_n, n > 1$.
Then which of the following options is/are correct?

Options:

(A) The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal

(B) The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$

(C) The length of latus rectum of $E_9$ is $\frac{1}{6}$

(D) $\sum_{n=1}^N (	ext{area of } R_n) < 24$, for each positive integer $N$

Sigma Percentile · Accepted Answer

### The Geometry of Infinity: Unraveling the Nested Ellipses

Welcome, students. Today, we are not just solving a problem; we are embarking on a journey into the heart of self-similarity. Imagine standing before a grand, infinite staircase of shapes—ellipses cradling rectangles, which in turn cradle smaller ellipses.

As we peel back the layers, you will see that the universe of this problem is governed by a beautiful, simple rhythm.

### Analyzing the Foundation ($E_1$ and $R_1$)

Let us start at the beginning. We are given $E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1$. By comparing this to the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, we immediately identify our starting parameters: $a_1 = 3$ and $b_1 = 2$.

Now, we must inscribe a rectangle $R_1$ of maximum area. To do this, we use the power of parametric coordinates. Let a vertex of the rectangle in the first quadrant be $P(a_1 \cos	heta, b_1 \sin	heta)$.

The rectangle, by symmetry, spans from $-x$ to $+x$ and $-y$ to $+y$. Thus, its dimensions are $2a_1 \cos	heta$ and $2b_1 \sin	heta$. The area $A$ is given by:

$$A = (2a_1 \cos	heta)(2b_1 \sin	heta) = 2a_1b_1 \sin(2	heta)$$

To maximize this, we need $\sin(2	heta)$ to be at its peak, which is $1$. This happens when $2	heta = \frac{\pi}{2}$, or $	heta = \frac{\pi}{4}$.

At this point, the area of $R_1$ is simply $2a_1b_1 = 2(3)(2) = 12$. We have successfully conquered the first step!

### The Recurrence Relation

Here is where the magic happens. The problem states that $E_2$ is the largest ellipse inscribed in $R_1$. For an ellipse to be inscribed in a rectangle, its semi-axes must be half the side lengths of the rectangle.

Since the vertices of $R_1$ are at $(\pm \frac{a_1}{\sqrt{2}}, \pm \frac{b_1}{\sqrt{2}})$, the semi-axes of $E_2$ are $a_2 = \frac{a_1}{\sqrt{2}}$ and $b_2 = \frac{b_1}{\sqrt{2}}$.

Do you see the pattern? Every time we move from $E_n$ to $E_{n+1}$, we scale the dimensions by $\frac{1}{\sqrt{2}}$. This is a geometric progression! For any $n$, the semi-axes are:

$$a_n = \frac{a_1}{(\sqrt{2})^{n-1}}, \quad b_n = \frac{b_1}{(\sqrt{2})^{n-1}}$$

### Evaluating the Truths

Now, let us test the options provided.

**1. The Eccentricity:** Option A suggests the eccentricities of $E_{18}$ and $E_{19}$ are not equal. But look at the formula $e_n = \sqrt{1 - \frac{b_n^2}{a_n^2}}$.

Since both $a_n$ and $b_n$ are scaled by the same factor, the ratio $\frac{b_n}{a_n}$ is constant. Thus, $e_n$ is constant for all $n$. Option A is incorrect.

**2. The Focus Distance:** For $E_9$, the distance of the focus from the center is $a_9 e$. We calculated $a_9 = \frac{a_1}{(\sqrt{2})^8} = \frac{3}{16}$.

With $e = \frac{\sqrt{5}}{3}$, the distance is $\frac{3}{16} 	imes \frac{\sqrt{5}}{3} = \frac{\sqrt{5}}{16}$. Option B claims it is $\frac{\sqrt{5}}{32}$, so it is incorrect.

**3. The Latus Rectum:** The length of the latus rectum is $\frac{2b_n^2}{a_n}$. For $E_9$, $b_9 = \frac{2}{(\sqrt{2})^8} = \frac{2}{16} = \frac{1}{8}$. Plugging these in:

$$LR = \frac{2(\frac{1}{8})^2}{\frac{3}{16}} = \frac{2(\frac{1}{64})}{\frac{3}{16}} = \frac{\frac{1}{32}}{\frac{3}{16}} = \frac{1}{6}$$

This matches Option C perfectly!

**4. The Sum of Areas:** Finally, the area of $R_n$ is $2a_n b_n = \frac{2a_1 b_1}{2^{n-1}} = \frac{12}{2^{n-1}}$. The sum of these areas is a geometric series: $12 + 6 + 3 + \dots$.

The sum of an infinite geometric series is $\frac{12}{1 - 0.5} = 24$. Since we are summing finite terms, the sum is strictly less than 24. Option D is correct.

### Conclusion

We have navigated the nested geometry and found that the latus rectum of $E_9$ is $\frac{1}{6}$ and the sum of the areas is bounded by 24. Remember, in JEE Advanced, the most complex-looking problems often hide the most elegant, simple patterns. Keep looking for that symmetry!

Select Answer:

Visualized Solution

Parameters of $E_{1}$

Area of Inscribed Rectangle $R_{1}$

Maximizing Area of $R_{1}$

Parameters of Ellipse $E_{2}$

Generalizing to $E_{n}$ and $R_{n}$

Checking Option A: Eccentricity

Checking Option B: Focus Distance of $E_{9}$

Checking Option C: Latus Rectum of $E_{9}$

Checking Option D: Area of $R_{n}$

Sum of Areas of Rectangles

Conclusion

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

The Geometry of Infinity

Analyzing the Foundation ( $E_{1}$ and $R_{1}$ )

The Recurrence Relation

Evaluating the Truths

Conclusion

Similar Questions

The ellipse $E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$ . The eccentricity of the ellipse $E_{2}$ is

Comprehension Passage

The orthocentre of the triangle $F_{1} M N$ is

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the x-axis at $Q$ , then the ratio of area of the triangle $M QR$ to area of the quadrilateral $M F_{1} N F_{2}$ is

Let the ellipse $E : \frac{x ^{2}}{144} + \frac{y ^{2}}{169} = 1$ and the hyperbola $H : \frac{x ^{2}}{16} - \frac{y ^{2}}{λ ^{2}} = - 1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$ , then the value of $24 (e + L)$ is :

Let $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ be an ellipse, whose eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $14$ . Then the square of the eccentricity of $\frac{x ^{2}}{a ^{2}} - \frac{m y ^{2}}{b ^{2}} = 1$ is :

Select Answer:

Visualized Solution

Parameters of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E1​

Area of Inscribed Rectangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​

Maximizing Area of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​

Parameters of Ellipse .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E2​

Generalizing to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } En​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Rn​

Checking Option A: Eccentricity

Checking Option B: Focus Distance of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E9​

Checking Option C: Latus Rectum of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E9​

Checking Option D: Area of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Rn​

Sum of Areas of Rectangles

Conclusion

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

The Geometry of Infinity

Analyzing the Foundation ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​)

The Recurrence Relation

Evaluating the Truths

Conclusion

Similar Questions

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocentre of the triangle F1​MN is

Parameters of $E_{1}$

Area of Inscribed Rectangle $R_{1}$

Maximizing Area of $R_{1}$

Parameters of Ellipse $E_{2}$

Generalizing to $E_{n}$ and $R_{n}$

Checking Option B: Focus Distance of $E_{9}$

Checking Option C: Latus Rectum of $E_{9}$

Checking Option D: Area of $R_{n}$

Analyzing the Foundation ( $E_{1}$ and $R_{1}$ )

The orthocentre of the triangle $F_{1} M N$ is