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 angle between the tangents drawn from a point $(-3, 2)$ to the ellipse $4x^2 + 9y^2 - 36 = 0$ is

If the straight line $x \cos \alpha + y \sin \alpha = p$ touches the curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,then prove that $a^{2} \cos^{2} \alpha + b^{2} \sin^{2} \alpha = p^{2}$.

The eccentricity of the conic $36x^2 + 144y^2 - 36x - 96y - 119 = 0$ is

If the normal drawn at one end of the latus rectum of the ellipse $b^2 x^2 + a^2 y^2 = a^2 b^2$ with eccentricity $e$ passes through one end of the minor axis,then:

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