Question

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 $x^2+(y-2{)}^2=4$ as its semi-major axis. If the centre of the ellipse is at the origin and its axes are the
coordinate axes, then the equation of the ellipse is

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

Answer 1

The correct option is D $x^2+4y^2=16$

Given:
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

Let
$(x-1{)}^2+y^2=1\to$ circle$1$
$x^2+(y-2{)}^2=4\to$ circle $2$

Semi- major axis of ellipse = Diameter of the circle $2$
Semi- minor axis of ellipse = Diameter of the circle $1$

From the equation of the circle $2$;
We get,
Radius = $2$
Diameter= $4$
$\therefore$ Semi-major axis = $4$

From the equation of the circle $1$;
We get,
Radius = $1$
Diameter= $2$
$\therefore$ Semi-minor axis = $2$

Note:General equation of an ellipse is given by,

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
where
$a$= semi-major axis
$b$= Semi-minor axis

From above,
$a$= diameter of circle $2$= $4$
$b$= diameter of circle $1$= $2$
$\therefore$ Equation of ellipse is
$\frac{x^2}{4^2}+\frac{y^2}{2^2}=1$
$\Rightarrow$ $\frac{x^2}{16}+\frac{y^2}{4}=1$
$\Rightarrow$ $x^2+4y^2=16$