The eccentricity of the ellipse $9x^2 + 5y^2 - 30y = 0$ is

If $x \cos \alpha + y \sin \alpha = 4$ is a tangent to $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$,then the value of $\alpha$ is

The equation of the locus of the foot of the perpendicular drawn from the centre of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to any tangent of the ellipse is

If the line $2x + 5y = 12$ intersects the ellipse $4x^2 + 5y^2 = 20$ in two distinct points $A$ and $B$,then the mid-point of $AB$ is

Let a tangent to the curve $9x^2 + 16y^2 = 144$ intersect the coordinate axes at the points $A$ and $B$. Then,the minimum length of the line segment $AB$ is $.........$

$A$ 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