The equation of the tangent to the ellipse $\frac{x^2}{4} + \frac{y^2}{12} = 1$ at the point $(1, 3)$ is:

Find the equation of the ellipse,whose length of the major axis is $20$ and foci are $(0, \pm 5)$.

If the line $\frac{x}{a} + \frac{y}{b} = \sqrt{2}$ is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then its eccentric angle $\theta = ............^{\circ}$

$A$ tangent is drawn at $(3 \sqrt{3} \cos \theta, \sin \theta)$ $\left(0 < \theta < \frac{\pi}{2}\right)$ to the ellipse $\frac{x^2}{27}+\frac{y^2}{1}=1$. The value of $\theta$ for which the sum of the intercepts on the coordinate axes made by this tangent attains the minimum is

The ellipse $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

The area of the quadrilateral formed by drawing tangents at the ends of the latus recta of the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ is