The equation of the ellipse whose one of the vertices is $(0, 7)$ and the corresponding directrix is $y = 12$ is:

If the equations $x = 1 + 2 \cos \theta$ and $y = 2 + \sin \theta$ for $0 \leq \theta < 2 \pi$ represent an ellipse,then the point of intersection of the normal drawn at $P(\theta = \pi/4)$ to this ellipse and its major axis is:

$A$ ray of light passing through $(2, 1)$ is reflected at a point $P$ on the $y$-axis and then passes through the point $(5, 3)$. If this reflected ray is the directrix of an ellipse with eccentricity $e = \frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}}$,then the equation of the other directrix can be:

The angle between the pair of tangents drawn from the point $(1, 2)$ to the ellipse $3x^2 + 2y^2 = 5$ is

In an ellipse with its centre at the origin,if the difference between the lengths of the major axis and the minor axis is $10$ and one of the foci is at $(0, 5\sqrt{3})$,then the length of its latus rectum is:

For which of the following curves is the line $x+\sqrt{3} y=2 \sqrt{3}$ a tangent at the point $\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right)$?