Key Concepts and Formulas

This problem requires a strong understanding of the standard form of an ellipse and its geometric relationship with inscribed and circumscribed rectangles, especially when all figures are centered at the origin and aligned with the coordinate axes.

1. Standard Equation of an Ellipse: An ellipse centered at the origin, with its major and minor axes lying along the coordinate axes, has the standard equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Here, $a$ represents the length of the semi-axis along the x-axis, and $b$ represents the length of the semi-axis along the y-axis.

   • The vertices along the x-axis are $(\pm a, 0)$.
   • The vertices along the y-axis are $(0, \pm b)$.

2. Ellipse Inscribed in a Rectangle (Aligned with Coordinate Axes): If an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is inscribed in a rectangle whose sides are parallel to the coordinate axes, it means the ellipse touches the midpoints of the rectangle's sides.

   • The points of tangency are $(\pm a, 0)$ and $(0, \pm b)$.
   • Consequently, the equations of the sides of this rectangle are $x = \pm a$ and $y = \pm b$.
   • The vertices of this rectangle are therefore $(\pm a, \pm b)$. This is the smallest possible rectangle that can contain the ellipse with its sides aligned with the axes.

3. Rectangle Inscribed in an Ellipse: If a rectangle with vertices $(\pm x_0, \pm y_0)$ is inscribed in an ellipse with equation $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$, it implies that all four vertices of the rectangle must lie on the ellipse.

   • Therefore, the coordinates of any one of these vertices (e.g., $(x_0, y_0)$) must satisfy the ellipse's equation: $\frac{x_0^2}{A^2} + \frac{y_0^2}{B^2} = 1$

---

Step-by-Step Solution

Step 1: Analyze the given first ellipse and determine its semi-axes.

The problem starts with the equation of an ellipse: $x^2 + 4y^2 = 4$ Why this step? To understand the dimensions and properties of this ellipse, we must convert its equation into the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. This form directly reveals the squares of its semi-axes.

To convert to standard form, divide the entire equation by the constant term on the right side, which is 4: $\frac{x^2}{4} + \frac{4y^2}{4} = \frac{4}{4}$ $\frac{x^2}{4} + \frac{y^2}{1} = 1$ From this standard form, we can identify the squares of the semi-axes for the first ellipse:

• $a_1^2 = 4 \implies a_1 = \sqrt{4} = 2$ (This is the length of the semi-axis along the x-axis for the first ellipse).
• $b_1^2 = 1 \implies b_1 = \sqrt{1} = 1$ (This is the length of the semi-axis along the y-axis for the first ellipse). We use subscripts ($a_1, b_1$) to clearly distinguish these parameters from those of the second ellipse we need to find.

Step 2: Determine the vertices of the rectangle.

The problem states that "The ellipse $x^2 + 4y^2 = 4$ is inscribed in a rectangle aligned with the coordinate axes." Why this step? The rectangle serves as a crucial link between the two ellipses. Its dimensions are directly determined by the first ellipse, and its vertices then define points that lie on the second ellipse.

According to our Key Concepts, if an ellipse $\frac{x^2}{a_1^2} + \frac{y^2}{b_1^2} = 1$ is inscribed in a rectangle aligned with the coordinate axes, the vertices of this rectangle are $(\pm a_1, \pm b_1)$. Using the values $a_1=2$ and $b_1=1$ from Step 1, the vertices of this rectangle are $(\pm 2, \pm 1)$. Due to symmetry, all four vertices lie on the second ellipse. We can choose any one of these vertices, for example, $P = (2, 1)$, to use in the next steps.

Step 3: Formulate the general equation of the second (circumscribing) ellipse.

The problem states that this rectangle "in turn is inscribed in another ellipse." Let this second ellipse be represented by the general equation: $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ Why this step? We need to find the specific equation of this new, outer ellipse. Since the rectangle is centered at the origin and its sides are aligned with the coordinate axes, the ellipse that circumscribes it must also be centered at the origin and have its axes aligned with the coordinate axes. We use $A^2$ and $B^2$ for its parameters to avoid confusion with $a_1^2$ and $b_1^2$ of the first ellipse. Our goal is to determine the values of $A^2$ and $B^2$.

Step 4: Use the given points to find the parameters ($A^2$ and $B^2$) of the circumscribing ellipse.

We have two pieces of information that will help us determine $A^2$ and $B^2$:

1. The second ellipse circumscribes the rectangle, meaning the rectangle's vertices lie on it. Thus, the point $P(2,1)$ (from Step 2) must satisfy its equation.
2. The second ellipse passes through the point $(4,0)$.

Using point $P(2,1)$ (a vertex of the inscribed rectangle): Substitute $x=2$ and $y=1$ into the general equation of the second ellipse $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$: $\frac{(2)^2}{A^2} + \frac{(1)^2}{B^2} = 1$ $\frac{4}{A^2} + \frac{1}{B^2} = 1 \quad \text{... (Equation i)}$ Why this substitution? Since the rectangle is inscribed in the second ellipse, its vertices must lie on the ellipse. Substituting the coordinates of one such vertex establishes a relationship between the unknown parameters $A^2$ and $B^2$.

Using point $(4,0)$ (a point the ellipse passes through): Substitute $x=4$ and $y=0$ into the general equation of the second ellipse: $\frac{(4)^2}{A^2} + \frac{(0)^2}{B^2} = 1$ $\frac{16}{A^2} + 0 = 1$ $\frac{16}{A^2} = 1$ $A^2 = 16$ Why this substitution? This point is explicitly given as lying on the second ellipse. Substituting it directly allows for a straightforward calculation of $A^2$ because the $y$-term becomes zero, simplifying the equation significantly and allowing us to find $A^2$ immediately.

Now, substitute the value of $A^2$ into Equation (i) to find $B^2$: Substitute $A^2 = 16$ into Equation (i): $\frac{4}{16} + \frac{1}{B^2} = 1$ $\frac{1}{4} + \frac{1}{B^2} = 1$ Why this step? We have now found the value for one parameter ($A^2$). By substituting this value into the equation that relates both parameters (Equation i), we can solve for the remaining unknown parameter, $B^2$.

Subtract $\frac{1}{4}$ from both sides of the equation: $\frac{1}{B^2} = 1 - \frac{1}{4}$ $\frac{1}{B^2} = \frac{4-1}{4}$ $\frac{1}{B^2} = \frac{3}{4}$ Inverting both sides gives us the value for $B^2$: $B^2 = \frac{4}{3}$

Step 5: Write the final equation of the circumscribing ellipse.

Now that we have determined $A^2 = 16$ and $B^2 = \frac{4}{3}$, we can substitute these values back into the general equation of the ellipse from Step 3: $\frac{x^2}{16} + \frac{y^2}{4/3} = 1$ Why this step? Having found both $A^2$ and $B^2$, we have all the necessary information to write the complete equation of the second ellipse. The final part of this step is to simplify the equation to match the format typically found in multiple-choice options, which usually involves integer coefficients.

To eliminate the denominators and express the equation with integer coefficients, we multiply the entire equation by the least common multiple (LCM) of the denominators, which are $16$ and $4/3$. The LCM of $16$ and $4/3$ is $16$. $16 \left( \frac{x^2}{16} \right) + 16 \left( \frac{y^2}{4/3} \right) = 16 \times 1$ $x^2 + \frac{16 \times 3y^2}{4} = 16$ $x^2 + (4 \times 3)y^2 = 16$ $x^2 + 12y^2 = 16$ Comparing this final equation with the given options, we find that it precisely matches option (A).

---

Tips and Common Mistakes

• Distinguish Parameters: When dealing with multiple ellipses or related geometric figures, always use distinct variables for their parameters (e.g., $a_1, b_1$ for the first ellipse and $A, B$ for the second). This prevents confusion and errors in calculations.
• Geometric Interpretation is Key: A clear understanding of the definitions of "inscribed in" and "circumscribed around" is paramount in this context.
  • An ellipse inscribed in a rectangle (aligned with axes) means the ellipse is tangent to the rectangle's sides at its vertices $(\pm a, 0)$ and $(0, \pm b)$. Therefore, the rectangle's vertices are $(\pm a, \pm b)$.
  • A rectangle inscribed in an ellipse means the rectangle's vertices lie on the ellipse.
• Algebraic Precision: Be extremely meticulous with algebraic manipulations, especially when dealing with fractions. A common mistake is miscalculating expressions like $\frac{16}{4/3}$. Remember that dividing by a fraction is equivalent to multiplying by its reciprocal: $\frac{16}{4/3} = 16 \times \frac{3}{4} = 12$.
• Symmetry Simplification: For figures centered at the origin and aligned with the coordinate axes, symmetry simplifies the problem significantly. You only need to consider one vertex of the rectangle (e.g., $(a,b)$) rather than all four, as they will all satisfy the equations due to the inherent symmetry.
• Final Form Check: Once you have derived the final equation, quickly compare it with the provided options. This can help confirm your result or sometimes highlight a small calculation error if your answer doesn't match any option.

---

Summary and Key Takeaway

This problem is an excellent illustration of how to combine the standard algebraic representation of an ellipse with its geometric properties. The key steps and takeaways are:

1. Standard Form Conversion: Always begin by converting any given ellipse equation into its standard form ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$) to easily extract its semi-axis lengths.
2. Translate Geometry to Algebra: Accurately translate geometric descriptions (such as "ellipse inscribed in a rectangle" or "rectangle inscribed in an ellipse") into algebraic coordinates (e.g., vertices $(\pm a, \pm b)$).
3. Systematic Parameter Determination: Utilize all given conditions (points known to be on the ellipse, vertices of inscribed figures) to form a system of equations. Solve this system systematically to determine the unknown parameters ($A^2, B^2$).
4. Equation Formulation and Simplification: Once all parameters are found, substitute them back into the general equation and simplify the resulting equation, typically to have integer coefficients, to match common answer formats.

This problem reinforces the importance of a systematic approach, careful algebraic handling, and a solid conceptual understanding of conic sections.