The equation of the chord of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$,whose mid-point is $(3, 1)$,is:

$A$ tangent is drawn to the ellipse $\frac{x^2}{27} + y^2 = 1$ at the point $(3\sqrt{3} \cos \theta, \sin \theta)$ (where $\theta \in (0, \frac{\pi}{2})$). Then,the value of $\theta$ such that the sum of the intercepts on the axes made by this tangent is minimum,is

The radius of a circle centered at $(0, 3)$ and passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

The eccentric angle of the point $P(-6, 2)$ on the ellipse $\frac{x^2}{48} + \frac{y^2}{16} = 1$ is: (in $^{\circ}$)

$A$ staircase of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the staircase,nearer to its end on the wall,that divides its length in the ratio $1 : 2$. If the staircase begins to slide on the floor,then the locus of $P$ is

Find the equations of the tangents to the ellipse $3x^{2} + 4y^{2} = 12$ which are perpendicular to the line $y + 2x = 4$.