Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5, 0)$,foci $(\pm 4, 0)$.

An ellipse with its minor and major axes parallel to the coordinate axes passes through $(0,0)$,$(1,0)$,and $(0,2)$. One of its foci lies on the $Y$-axis. The eccentricity of the ellipse is

The eccentricity of an ellipse,with its centre at the origin,is $\frac{1}{2}$. If one of the directrices is $x = 4$,then the equation of the ellipse is

The equation of the normal drawn at the point $(\sqrt{2}+1, -1)$ to the ellipse $x^2+2y^2-2x+8y+5=0$ is

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