If the distance between the foci of an ellipse is $6$ and the length of the minor axis is $8$,then the eccentricity 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

The maximum length of a chord of the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$,such that the eccentric angles of its extremities differ by $\frac{\pi}{2}$,is:

The curve represented by $\frac{x^2}{12-\alpha} + \frac{y^2}{\alpha-10} = 1$ is

Find the area of the quadrilateral formed by the tangents at the endpoints of the latus rectum of the ellipse $x^2 + 2y^2 = 2$.

Let the line $y=mx$ and the ellipse $2x^{2}+y^{2}=1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the coordinate axes at $(-\frac{1}{3\sqrt{2}}, 0)$ and $(0, \beta)$,then $\beta$ is equal to