The mid-point of a chord of the ellipse $x^2+4y^2-2x+20y=0$ is $(2,-4)$. The equation of the chord is

The point $(4, -3)$ with respect to the ellipse $4x^2 + 5y^2 = 1$ is:

If the line $x - 2y = 12$ is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(3, -4.5)$,then the length of the latus rectum of the ellipse is:

Let $S$ and $S^{\prime}$ be the foci of an ellipse and $B$ be one end of its minor axis. If $\triangle SBS^{\prime}$ is an isosceles right-angled triangle,then the eccentricity of the ellipse is

The lengths of the sides of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$ are

If the area of the auxiliary circle of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (where $a > b$) is twice the area of the ellipse,then the eccentricity of the ellipse is