The equation of the ellipse referred to its axes as the axes of coordinates with latus rectum of length $4$ and distance between foci $4 \sqrt 2$ is-

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point  $(-3,1) $ and has eccentricity $\sqrt {\frac{2}{5}} $ is 

Statement $-1$ : If two tangents are drawn to an ellipse from a single point and if they are perpendicular to each other, then locus of that point is always a circle 

Statement $-2$ : For an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ , locus of that point from which two perpendicular tangents are drawn, is $x^2 + y^2 = (a + b)^2$ .

The equation of the ellipse whose vertices are $( \pm 5,\;0)$ and foci are $( \pm 4,\;0)$ is

The equation of the ellipse whose latus rectum is $8$ and whose eccentricity is $\frac{1}{{\sqrt 2 }}$, referred to the principal axes of coordinates, is

The product of the lengths of perpendiculars from the foci on any tangent to the ellipse $3x^2 + 5y^2 = 1$, is