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:

Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$,pass through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and have eccentricity $e = \frac{1}{\sqrt{3}}$. If a circle,centered at the focus $F(\alpha, 0), \alpha > 0$ of $E$ and having radius $r = \frac{2}{\sqrt{3}}$,intersects $E$ at two points $P$ and $Q$,then $PQ^{2}$ is equal to:

The line $y = 2t^2$ intersects the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ in real points if

The locus of the midpoints of the intercepted portion of the tangents by the coordinate axes,which are drawn to the ellipse $x^2+2y^2=2$,is

The midpoint of the chord of the ellipse $x^2 + \frac{y^2}{4} = 1$ formed on the line $y = x + 1$ is

$A$ particle is travelling in a clockwise direction on the ellipse $\frac{x^2}{100} + \frac{y^2}{25} = 1$. If the particle leaves the ellipse at the point $(-8, 3)$ and travels along the tangent to the ellipse at that point,then the point where the particle crosses the $Y$-axis is: