1. Find the semi-minor axis length

The first circle is

Its radius is $1$, so its diameter is

This diameter is given as the semi-minor axis of the ellipse.

Hence,

where $b$ is the semi-minor axis.

2. Find the semi-major axis length

The second circle is

Its radius is $2$, so its diameter is

This diameter is given as the semi-major axis of the ellipse.

Hence,

where $a$ is the semi-major axis.

3. Form the equation of the ellipse

The ellipse is centered at the origin and its axes are the coordinate axes.

Since $a>b$, the major axis must lie along one coordinate axis and minor axis along the other. The standard forms are:

• Major axis along $x$-axis: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
• Major axis along $y$-axis: $$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$$

Here the options suggest the major axis is along the $x$-axis, so substitute

Then

Multiplying by $16$,

4. Check options

• A: $4x^2+y^2=4$ gives semi-axes $1,2$ → not correct.
• B: $x^2+4y^2=8$ gives semi-axes $2\sqrt2,\sqrt2$ → not correct.
• C: $4x^2+y^2=8$ gives semi-axes $\sqrt2,2\sqrt2$ → not correct.
• D: $x^2+4y^2=16$ gives semi-axes $4,2$ → correct.

So the correct option is