How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2, 3)$?

$L_1^{\prime}$ is the end of a latus rectum of the ellipse $3x^2 + 4y^2 = 12$ which is lying in the third quadrant. If the normal drawn at $L_1^{\prime}$ to this ellipse intersects the ellipse again at the point $P(a, b)$,then $a =$

The equation of the ellipse whose foci are $(\pm 5, 0)$ and one of its directrices is $5x = 36$ is:

For the ellipse $4x^2 + y^2 - 8x + 2y + 1 = 0$,the focus and the equation of the directrix are respectively

If a focal chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ meets its minor axis at the point $(0,3)$,then the perpendicular distance from the centre of the ellipse to this focal chord is

The equation of a chord $AB$ of an ellipse $2x^2 + y^2 = 1$ is $x - y + 1 = 0$. If $O$ is the origin,then $\angle AOB =$