The foci of the ellipse $25x^2 + 4y^2 + 100x - 4y + 100 = 0$ are

The line $x = 8$ is the directrix of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with the corresponding focus $(2, 0)$. If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $(0, 4\sqrt{3})$ and intersects the $x$-axis at $Q$,then $(3PQ)^2$ is equal to $........$

If a circle $(x-1)^2+y^2=r^2$ touches the ellipse $x^2+4y^2=16$ internally,then $r=$

Find the number of points on the ellipse $\frac{x^{2}}{50} + \frac{y^{2}}{20} = 1$ from which a pair of perpendicular tangents can be drawn to the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$.

For the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$,a chord $PQ$ subtends a right angle at the center. What is the locus of the point of intersection of the tangents at $P$ and $Q$?

If $z$ is a complex number in the Argand plane,then the equation $|z - 2| + |z + 2| = 8$ represents