Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac{1}{2}$. If $P(1, \beta), \beta>0$ is a point on this ellipse,then the equation of the normal to it at $P$ is

The eccentricity of the ellipse $9x^2 + 5y^2 - 18x - 2y - 16 = 0$ is

If $\frac{\pi}{3}$ and $\theta$ are the eccentric angles of the ends of a focal chord of the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$,then $\tan \theta=$

Let $F_1$ and $F_2$ be the foci of an ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$. $A$ ray from $F_1$ strikes the elliptical mirror at point $P$ and is reflected. What is the equation of the angle bisector of the angle between the incident ray and the reflected ray?

$A$ tangent drawn at a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cuts the $X$-axis at point $A$. If $A^{\prime}$ is the image of $A$ with respect to the line $y=x$,then the circle with $AA^{\prime}$ as its diameter passes through the fixed point:

If $A_1, A_2, A_3$ are the areas of the ellipse $x^2+4y^2-4=0$, its director circle, and its auxiliary circle respectively, then $A_2+A_3-A_1=$ (in $\pi$)