The equation of the ellipse in the standard form whose length of the latus rectum is $4$ and whose distance between the foci is $4 \sqrt{2}$,is

For the ellipse $\frac{x^2}{64} + \frac{y^2}{28} = 1$,the eccentricity is

Statement $-1$: If two tangents are drawn to an ellipse from a single point and if they are perpendicular to each other,then the locus of that point is always a circle.
Statement $-2$: For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,the locus of the point from which two perpendicular tangents are drawn is $x^2 + y^2 = a^2 + b^2$.

The eccentricity of the ellipse $9x^2 + 25y^2 = 225$ is

$A$ tangent drawn at a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cuts the $X$-axis at point $A$. If $A^{\prime}$ is the image of $A$ with respect to the line $y=x$,then the circle with $AA^{\prime}$ as its diameter passes through the fixed point:

The equation of the ellipse whose centre is $(2, -3)$,one of the foci is $(3, -3)$ and the corresponding vertex is $(4, -3)$ is