Question

An ellipse is drawn by taking the diameter of the circle ${(x-1)}^2+y^2=1$as its semi-minor axis and the diameter of the circle$x^2+{(y-2)}^2=4$ as the semi-major axis, if the center of the ellipse at the origin and its axis are the coordinates axes, then the equation of the ellipse is

• A. $4x^2+y^2=4$
• B. $x^2+4y^2=8$
• C. $4x^2+y^2=8$
• D. $x^2+4y^2=16$

Answer 1

The correct option is D

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

Explanation for the correct option:

Find the equation of the ellipse :

Given the circle${(x-1)}^2+y^2=1$

Radius $=1$ and diameter $=2$

$\therefore$ Length of semi-minor axis $=$diameter of the circle ${(x-1)}^2+y^2=1$(given)

$\therefore$ $b=2$

And Circle $x^2+{(y-2)}^2=4$

Radius $=2$ and diameter $=4$

$\therefore$ length of the semi-major axis $=$diameter of the circle $x^2+{(y-2)}^2=4$ (given)

$\therefore$ $a=4$

Therefore the equation of the ellipse

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

Hence the correct option is D.