If $A = [(x,\,y):{x^2} + {y^2} = 25]$ and $B = [(x,\,y):{x^2} + 9{y^2} = 144]$, then $A \cap B$ contains

Let $L$ be a common tangent line to the curves $4 x^{2}+9 y^{2}=36$ and $(2 x)^{2}+(2 y)^{2}=31$. Then the square of the slope of the line $L$ is ..... .

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{36}+\frac{y^2} {16}=1$

Consider an ellipse with foci at $(5,15)$ and $(21,15)$. If the $X$-axis is a tangent to the ellipse, then the length of its major axis equals

Consider ellipses $E _{ k }: kx ^2+ k ^2 y ^2=1, k =1,2, \ldots$,$20$. Let $C _{ k }$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$, If $r_k$ is the radius of the circle $C _{ k }$, then the value of $\sum \limits_{ k =1}^{20} \frac{1}{ I _{ k }^2}$ is $.......$.

The equation of the tangent to the ellipse ${x^2} + 16{y^2} = 16$ making an angle of ${60^o}$ with $x$ - axis is