An ellipse is drawn by taking a diameter of the circle ${\left( {x - 1} \right)^2} + {y^2} = 1$ as its semi-minor axis and a diameter of the circle ${x^2} + {\left( {y - 2} \right)^2} = 4$ is semi-major axis. If the center of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :

Let $A = \left\{ {\left( {x,y} \right):\,y = mx + 1} \right\}$ 

      $B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$ 

$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ . 

If set $C$ is singleton set then sum of all possible values of $m$ is

If the eccentricity of an ellipse be $1/\sqrt 2 $, then its latus rectum is equal to its

An ellipse passes through the point $(-3, 1)$ and its eccentricity is $\sqrt {\frac{2}{5}} $. The equation of the ellipse is

The length of the latus rectum of the ellipse $9{x^2} + 4{y^2} = 1$, is

The equation of the ellipse referred to its axes as the axes of coordinates with latus rectum of length $4$ and distance between foci $4 \sqrt 2$ is-