The equation of the normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(a \cos \theta, b \sin \theta)$ is

Let the ellipse $E : x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A, P$ and the origin $O$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $m - n$ is equal to

Let the equation of an ellipse be $\frac{x^{2}}{144}+\frac{y^{2}}{25}=1$. Then, the radius of the circle with center $(0, \sqrt{2})$ and passing through the foci of the ellipse is

The length of the latus rectum of an ellipse is $6$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{5}{3}$ units. If $e$ is the eccentricity of this ellipse,then $e$ satisfies the equation:

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 number of tangents to the circle $x^2 + y^2 = 3$ that are normal to the ellipse $4x^2 + 9y^2 = 36$ is