Understanding the Problem

We are given two circles and need to form an ellipse with its center at the origin (0,0) and axes along the coordinate axes. The semi-minor axis (b) is a diameter of the first circle, and the semi-major axis (a) is a diameter of the second circle. We must find the correct equation of this ellipse from the given options.

The standard equation of an ellipse centered at the origin with axes along the coordinate axes is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where a is the semi-major axis and b is the semi-minor axis (a > b).

Step-by-Step Solution

Step 1: Find the Length of the Semi-Minor Axis (b)

The semi-minor axis is a diameter of the circle: ${(x-1)}^2+y^2=1$.

The standard form of a circle is ${(x-h)}^2+{(y-k)}^2=r^2$, where (h,k) is the center and r is the radius.

For our first circle: Center = (1, 0), Radius (r₁) = $\sqrt{1}=1$.

The length of a diameter is twice the radius. Therefore, the length of the semi-minor axis b is equal to the diameter of this circle.

$b=2\times r_1=2\times 1=2$

So, $b^2=2^2=4$.

Step 2: Find the Length of the Semi-Major Axis (a)

The semi-major axis is a diameter of the circle: $x^2+{(y-2)}^2=4$.

For this second circle: Center = (0, 2), Radius (r₂) = $\sqrt{4}=2$.

The length of the semi-major axis a is equal to the diameter of this circle.

$a=2\times r_2=2\times 2=4$

So, $a^2=4^2=16$.

Step 3: Write the Equation of the Ellipse

We have a = 4 and b = 2. Since a > b, the major axis is along the x-axis.

Substituting the values into the standard ellipse equation:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{x^2}{16}+\frac{y^2}{4}=1$

Step 4: Compare with the Given Options

To match the format of the options, let's eliminate the denominators by multiplying the entire equation by the Least Common Multiple (LCM) of 16 and 4, which is 16.

$16\times (\frac{x^2}{16}+\frac{y^2}{4})=16\times 1$

$x^2+4y^2=16$

This matches the second option: x² + 4y² = 16.

Final Answer

The equation of the ellipse is $x^2+4y^2=16$.

Related Topics & Formulae

1. Standard Equation of a Circle

${(x-h)}^2+{(y-k)}^2=r^2$

Where (h, k) is the center and r is the radius.

2. Standard Equation of an Ellipse

Center at Origin, Major Axis horizontal (along x-axis):

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where a > b.

Center at Origin, Major Axis vertical (along y-axis):

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, where a > b.

In both cases, 'a' is always the semi-major axis and 'b' is the semi-minor axis.

3. Key Properties

• Diameter of a circle: A straight line passing from side to side through the center. Its length is 2r.
• Major vs. Minor Axis: The longer diameter of an ellipse is the major axis; the shorter one is the minor axis.