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

If $3x + 4y = 12\sqrt{2}$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{9} = 1$ for some $a \in R$,then the distance between the foci of the ellipse is:

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$ meet at the point $R(\sqrt{2}, 2\sqrt{2}-2)$. If $S$ is the focus of the ellipse on its negative major axis,then $SP^{2} + SQ^{2}$ is equal to.

If the eccentricity and the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are $\frac{\sqrt{3}}{2}$ and $1$ respectively,then the sum of the lengths of the major axis and minor axis of the ellipse is

The equation of the auxiliary circle of the ellipse $16 x^{2}+25 y^{2}+32 x-100 y=284$ is

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the latus rectum of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$.