Key Concept: Standard Equation of an Ellipse

An ellipse centered at the origin $(0,0)$ with its major and minor axes along the coordinate axes has a standard equation. Let $a$ be the length of the semi-major axis and $b$ be the length of the semi-minor axis. By definition, for an ellipse, the semi-major axis is always longer than the semi-minor axis, i.e., $a > b$.

There are two standard forms based on the orientation of the major axis:

1. If the major axis is along the x-axis: The equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ In this case, the vertices are $(\pm a, 0)$ and co-vertices are $(0, \pm b)$.
2. If the major axis is along the y-axis: The equation is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ In this case, the vertices are $(0, \pm a)$ and co-vertices are $(\pm b, 0)$.

To solve this problem, we need to determine the lengths of the semi-major axis ($a$) and the semi-minor axis ($b$) by analyzing the given circles. Then, we will use these lengths to write the equation of the ellipse.

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Step 1: Determine the length of the semi-minor axis ($b$)

The problem states that a diameter of the circle $(x - 1)^2 + y^2 = 1$ is taken as the semi-minor axis of the ellipse.

• Understanding the circle equation: The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Applying to the first circle: For the circle $(x - 1)^2 + y^2 = 1$, we can see that $r^2 = 1$.
• Calculating the radius: Therefore, the radius $r_1 = \sqrt{1} = 1$.
• Calculating the diameter: The diameter of this circle is $D_1 = 2 \times r_1 = 2 \times 1 = 2$.
• Assigning to semi-minor axis: Since this diameter is the semi-minor axis of the ellipse, we have $b = D_1 = 2$. So, $b = 2$.

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Step 2: Determine the length of the semi-major axis ($a$)

The problem states that a diameter of the circle $x^2 + (y - 2)^2 = 4$ is taken as the semi-major axis of the ellipse.

• Applying to the second circle: For the circle $x^2 + (y - 2)^2 = 4$, we can see that $r^2 = 4$.
• Calculating the radius: Therefore, the radius $r_2 = \sqrt{4} = 2$.
• Calculating the diameter: The diameter of this circle is $D_2 = 2 \times r_2 = 2 \times 2 = 4$.
• Assigning to semi-major axis: Since this diameter is the semi-major axis of the ellipse, we have $a = D_2 = 4$. So, $a = 4$.

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Step 3: Determine the orientation of the ellipse

We have found the lengths of the semi-major and semi-minor axes:

• Semi-major axis length, $a = 4$.

• Semi-minor axis length, $b = 2$.

• Verify condition: As expected for an ellipse, $a > b$ (i.e., $4 > 2$).

• Identify major axis: The problem statement explicitly says "a diameter of the circle $x^2 + (y-2)^2 = 4$ is semi-major axis". This circle has its center on the y-axis (at $(0,2)$), which might intuitively suggest the major axis is along the y-axis, but this is a common trap. The length of the diameter is $a$, not its orientation.

• Crucial Information: The problem states: "If the centre of the ellipse is at the origin and its axes are the coordinate axes". It does not specify whether the major axis is along x or y. We must determine this from the context of how $a$ and $b$ were defined or if there's any implicit information.

  • The problem defined $a$ as the semi-major axis and $b$ as the semi-minor axis.
  • The problem then asks for "the equation of the ellipse". We need to deduce the orientation.
  • Wait, let's re-read carefully: "An ellipse is drawn by taking a diameter of the circle $(x-1)^2 + y^2 = 1$ as its semi-minor axis and a diameter of the circle $x^2 + (y-2)^2 = 4$ is semi-major axis." This tells us which length corresponds to $a$ and $b$.
  • The key is in the options and the standard form: We have $a=4$ and $b=2$.
    • If the major axis is along the x-axis, the equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{x^2}{4^2} + \frac{y^2}{2^2} = 1 \implies \frac{x^2}{16} + \frac{y^2}{4} = 1$.
    • If the major axis is along the y-axis, the equation is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \implies \frac{x^2}{2^2} + \frac{y^2}{4^2} = 1 \implies \frac{x^2}{4} + \frac{y^2}{16} = 1$.

  Let's check the given options: (A) $4x^2 + y^2 = 4 \implies \frac{4x^2}{4} + \frac{y^2}{4} = \frac{4}{4} \implies \frac{x^2}{1} + \frac{y^2}{4} = 1$. (B) $x^2 + 4y^2 = 8 \implies \frac{x^2}{8} + \frac{4y^2}{8} = \frac{8}{8} \implies \frac{x^2}{8} + \frac{y^2}{2} = 1$. (C) $4x^2 + y^2 = 8 \implies \frac{4x^2}{8} + \frac{y^2}{8} = \frac{8}{8} \implies \frac{x^2}{2} + \frac{y^2}{8} = 1$. (D) $x^2 + 4y^2 = 16 \implies \frac{x^2}{16} + \frac{4y^2}{16} = \frac{16}{16} \implies \frac{x^2}{16} + \frac{y^2}{4} = 1$.

  Comparing our derived forms with the options:

  • Our first possibility: $\frac{x^2}{16} + \frac{y^2}{4} = 1$. This matches option (D).
  • Our second possibility: $\frac{x^2}{4} + \frac{y^2}{16} = 1$. This matches option (A).

  This means we need to definitively determine the orientation. The problem states "its axes are the coordinate axes" but doesn't explicitly state which is major. Let's re-evaluate the problem statement very carefully. "An ellipse is drawn by taking a diameter of the circle $(x-1)^2 + y^2 = 1$ as its semi-minor axis and a diameter of the circle $x^2 + (y-2)^2 = 4$ is semi-major axis." This phrasing defines the lengths $a$ and $b$. It does not define which coordinate axis corresponds to the major or minor axis. This implies that the problem expects us to infer the orientation from the final equation and options. However, the standard convention is that $a$ is always the length associated with the variable under the larger denominator in the standard form.

  Let's assume the question implies the axis corresponding to the semi-major axis is the one with length $a$. If $a=4$ and $b=2$:

  • If the major axis is along the x-axis, then $a^2=16$ is under $x^2$, and $b^2=4$ is under $y^2$. Equation: $\frac{x^2}{16} + \frac{y^2}{4} = 1$.
  • If the major axis is along the y-axis, then $a^2=16$ is under $y^2$, and $b^2=4$ is under $x^2$. Equation: $\frac{x^2}{4} + \frac{y^2}{16} = 1$.

  Both of these equations are valid possibilities based on the given lengths. We need to check which one matches an option. Option (A) is $\frac{x^2}{1} + \frac{y^2}{4} = 1$. Here, $b^2=1$ and $a^2=4$, so $b=1, a=2$. This doesn't match our calculated $a=4, b=2$. Option (D) is $\frac{x^2}{16} + \frac{y^2}{4} = 1$. Here, $a^2=16$ and $b^2=4$, so $a=4, b=2$. This exactly matches our calculated lengths and implies the major axis is along the x-axis.

  Wait, the correct answer is (A). This indicates my interpretation of options (A) and (D) is flawed or there's a misunderstanding.

  Let's re-examine the options after getting the values for $a$ and $b$. $a=4$, $b=2$.

  Case 1: Major axis along x-axis. Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{x^2}{4^2} + \frac{y^2}{2^2} = 1 \implies \frac{x^2}{16} + \frac{y^2}{4} = 1$. To convert to integer coefficients: Multiply by 16. $x^2 + 4y^2 = 16$. This is Option (D).

  Case 2: Major axis along y-axis. Equation: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \implies \frac{x^2}{2^2} + \frac{y^2}{4^2} = 1 \implies \frac{x^2}{4} + \frac{y^2}{16} = 1$. To convert to integer coefficients: Multiply by 16. $4x^2 + y^2 = 16$. This is not among the options.

  The correct answer is given as (A) $4x^2 + y^2 = 4$. Let's convert option (A) to standard form: $4x^2 + y^2 = 4 \implies \frac{4x^2}{4} + \frac{y^2}{4} = \frac{4}{4} \implies \frac{x^2}{1} + \frac{y^2}{4} = 1$. For this equation: The denominator under $x^2$ is $1$. The denominator under $y^2$ is $4$. Since $4 > 1$, the major axis is along the y-axis. So, $a^2 = 4 \implies a = 2$. And $b^2 = 1 \implies b = 1$.

  This means if option (A) is correct, then the semi-major axis length $