The area of the region enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is . . . . . . square units. (in $\pi$)

If $\theta$ is the eccentric angle of a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ such that the distance of the point from the center is $5$, then $\theta = ......$

The lines $y=2x+\sqrt{76}$ and $2y+x=8$ touch the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$. If the point of intersection of these two lines lies on a circle whose centre coincides with the centre of that ellipse,then the equation of that circle is

The center of the ellipse $x^2+2y^2-4x+12y+14=0$ is

For what value of $c$ does the line $y = 4x + c$ touch the curve $\frac{x^2}{4} + y^2 = 1$? Find the number of possible values of $c$.

The equation of the ellipse with directrix $3x+4y-5=0$,focus $(1,2)$ and eccentricity $e = \frac{1}{2}$,is