From point $P(8, 27)$,tangents $PQ$ and $PR$ are drawn to the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$. The angle subtended by $QR$ at the origin is:

The length of the latus rectum of $9x^2 + 25y^2 - 90x - 150y + 225 = 0$ is

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$,then the length of its latus rectum is

If the normal at an end of a latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through an extremity of the minor axis,then the eccentricity $e$ of the ellipse satisfies:

Let the common tangents to the curves $4(x^{2}+y^{2}) = 9$ and $y^{2} = 4x$ intersect at the point $Q$. Let an ellipse,centered at the origin $O$,have lengths of semi-minor and semi-major axes equal to $OQ$ and $6$,respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse,then $\frac{l}{e^{2}}$ is equal to

The normal at a point $P$ on the ellipse $x^2 + 4y^2 = 16$ meets the $x$-axis at $Q$. If $M$ is the midpoint of the line segment $PQ$,then the locus of $M$ intersects the latus rectum of the given ellipse at which points?