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(A) The given circle is $(x - 1)^2 + y^2 = 1$. Its radius is $r_1 = 1$,so its diameter is $d_1 = 2$. This is the semi-minor axis length,$b = 2$.The se

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

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(A) The given circle is $(x - 1)^2 + y^2 = 1$. Its radius is $r_1 = 1$,so its diameter is $d_1 = 2$. This is the semi-minor axis length,$b = 2$.The second circle is $x^2 + (y - 2)^2 = 4$. Its radius is $r_2 = 2$,so its diameter is $d_2 = 4$. This is the semi-major axis length,$a = 4$.The standard equation of an ellipse with center at origin and axes as coordinate axes is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (assuming major axis along $x$-axis) or $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (assuming major axis along $y$-axis).Case $1$: If major axis is along $y$-axis,$a = 4$ and $b = 2$. The equation is $\frac{x^2}{2^2} + \frac{y^2}{4^2} = 1$ $\Rightarrow \frac{x^2}{4} + \frac{y^2}{16} = 1$ $\Rightarrow 4x^2 + y^2 = 16$.Case $2$: If major axis is along $x$-axis,$a = 4$ and $b = 2$. The equation is $\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1$ $\Rightarrow \frac{x^2}{16} + \frac{y^2}{4} = 1$ $\Rightarrow x^2 + 4y^2 = 16$.Comparing with the given options,$4x^2 + y^2 = 16$ is present in option $A$.

An ellipse is drawn such that the diameter of the circle $(x - 1)^2 + y^2 = 1$ is the semi-minor axis and the diameter of the circle $x^2 + (y - 2)^2 = 4$ is the semi-major axis. If the center of the ellipse is at the origin and its axes are the coordinate axes,find the equation of the ellipse.

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Find the equation of the ellipse $(a > b)$ whose distance between the foci is $8$ and the distance between the directrices is $18$.

Statement $I$: The equation of the directrix of the ellipse $4x^2+y^2-8x-4y+4=0$ is $3y=6-4\sqrt{3}$.
Statement $II$: The equation of the latus rectum of the ellipse $x^2+4y^2-4x-8y+4=0$ is $y=2+\sqrt{3}$.
Which of the above statement$(s)$ is (are) true?

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