Let E1 : x²/a² + y²/b² = 1, a b. Let E2 be another ellipse such that it touches the end points of

Q: Wait, why is $E_2$ a vertical ellipse? The problem doesn't explicitly say so.

Great observation! The problem states that the foci of $E_2$ are the endpoints of the minor axis of $E_1$. The minor axis endpoints of $E_1$ are $(0, \pm b)$. Since these points lie on the y-axis, the foci of $E_2$ must lie on the y-axis. By definition, the major axis of an ellipse contains its foci. Therefore, the major axis of $E_2$ must be along the y-axis, making it a vertical ellipse.

Q: Why do we reject the negative root when solving the quadratic equation for $e$?

Eccentricity $e$ is defined as the ratio of the distance between the foci to the length of the major axis. By definition, this distance and length are positive, so $e$ must be greater than or equal to zero. In the context of an ellipse, $0 \le e < 1$. Since our quadratic solution yielded a negative value, it is physically impossible for an ellipse, so we discard it.

Q: Can we use the standard equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ for $E_2$ as well?

You can, but you must be careful with the labels! For $E_2$, the major axis is vertical, so the equation is $\frac{x^2}{a^2} + \frac{y^2}{c^2} = 1$ where $c > a$. If you use the standard form, just ensure that the denominator under the $y^2$ term is larger than the denominator under the $x^2$ term. My derivation used $c$ as the semi-major axis to avoid confusion with the $b$ from $E_1$.

Question

Let $E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$. Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1$. If $E_1$ and $E_2$ have same eccentricities, then its value is :

Options:

(A) $\frac{-1+\sqrt{5}}{2}$

(B) $\frac{-1+\sqrt{8}}{2}$

(C) $\frac{-1+\sqrt{3}}{2}$

(D) $\frac{-1+\sqrt{6}}{2}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane, looking at two beautiful, interconnected ellipses. The first, $E_1$, is a classic horizontal ellipse, stretched comfortably along the x-axis. Its soul is defined by its semi-major axis $a$ and semi-minor axis $b$.

The second, $E_2$, is a bit more mysterious. It is a vertical ellipse that dances with $E_1$, touching it at the very tips of its major axis, $(\pm a, 0)$. This is not just a geometry problem; it is a study in how two distinct shapes can share the same fundamental DNA—their eccentricity $e$.

### Decoding $E_1$

Let us start with $E_1$. The equation is:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

The eccentricity $e$ is the measure of how 'flat' the ellipse is. We know the relationship $e^2 = 1 - \frac{b^2}{a^2}$.

If we rearrange this, we get a very useful bridge:

$$ \frac{b^2}{a^2} = 1 - e^2 $$

Keep this in your pocket; we will need it later.

### The Vertical Mystery of $E_2$

Now, look at $E_2$. The problem states its foci are the endpoints of the minor axis of $E_1$, which are $(0, \pm b)$. Because the foci are on the y-axis, $E_2$ must be a vertical ellipse.

It touches $E_1$ at $(\pm a, 0)$, so its horizontal width is $2a$. This means the semi-minor axis of $E_2$ is $a$. Let us call its semi-major axis $c$.

Since it is a vertical ellipse, its equation is:

$$ \frac{x^2}{a^2} + \frac{y^2}{c^2} = 1 $$

The eccentricity relation for this vertical ellipse is $e^2 = 1 - \frac{a^2}{c^2}$. Rearranging this gives:

$$ \frac{a^2}{c^2} = 1 - e^2 \quad 	ext{or} \quad \frac{c^2}{a^2} = \frac{1}{1 - e^2} $$

### The Algebraic Bridge

We have two expressions for the eccentricity, but we need to connect them. We know the foci of $E_2$ are at $(0, \pm c \cdot e)$. But the problem says they are at $(0, \pm b)$.

Therefore, $c \cdot e = b$, which means $c = \frac{b}{e}$. Now, substitute this into our expression for $E_2$:

$$ \frac{c^2}{a^2} = \frac{(b/e)^2}{a^2} = \frac{b^2}{a^2 \cdot e^2} $$

We also know from the eccentricity of $E_2$ that $\frac{a^2}{c^2} = 1 - e^2$, so $\frac{c^2}{a^2} = \frac{1}{1 - e^2}$. Equating these two expressions for $\frac{c^2}{a^2}$, we get:

$$ \frac{b^2}{a^2 \cdot e^2} = \frac{1}{1 - e^2} $$

This simplifies to:

$$ \frac{b^2}{a^2} = \frac{e^2}{1 - e^2} $$

### The Final Convergence

We now have two different expressions for $\frac{b^2}{a^2}$. From $E_1$, we have $\frac{b^2}{a^2} = 1 - e^2$. From $E_2$, we have $\frac{b^2}{a^2} = \frac{e^2}{1 - e^2}$.

Set them equal:

$$ 1 - e^2 = \frac{e^2}{1 - e^2} $$

Cross-multiply to get $(1 - e^2)^2 = e^2$. Taking the square root of both sides, and remembering that $e < 1$, we get $1 - e^2 = e$.

This leads us to the quadratic equation $e^2 + e - 1 = 0$. Using the quadratic formula:

$$ e = \frac{-1 \pm \sqrt{1 - 4(1)(-1)}}{2} = \frac{-1 \pm \sqrt{5}}{2} $$

Since $e$ must be positive, we take the positive root:

$$ \mathbf{e = \frac{\sqrt{5} - 1}{2}} $$

This is the conjugate of the golden ratio! It is a beautiful, elegant result that emerges from the simple interaction of two ellipses.

Select Answer:

Visualized Solution

Analyze Ellipse $E_{1}$

Eccentricity of $E_{1}$

Defining Ellipse $E_{2}$

Foci of $E_{2}$

Parameters of $E_{2}$

Foci Relation for $E_{2}$

Eccentricity Relation for $E_{2}$

Combining $E_{2}$ Equations

Equating Expressions for $\frac{b ^{2}}{a ^{2}}$

Forming the Quadratic Equation

Solving for $e$

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

Analyzing the Setup

Decoding $E_{1}$

The Vertical Mystery of $E_{2}$

The Algebraic Bridge

The Final Convergence

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Select Answer:

Visualized Solution

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

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

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

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

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

Foci Relation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E2​

Eccentricity Relation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E2​

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

Equating Expressions for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2b2​