For the ellipse $25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$ the eccentricity $e = $

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 :

If the chord through the point whose eccentric angles are $\theta \,\& \,\phi $ on the ellipse,$(x^2/a^2) + (y^2/b^2) = 1$  passes through the focus, then the value of $ (1 + e)$ $\tan(\theta /2) \tan(\phi /2)$ is

If $OB$ is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity of the ellipse is

For the ellipse $3{x^2} + 4{y^2} = 12$, the length of latus rectum is

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $e$ of the ellipse satisfies