The length of the latus rectum of the ellipse $5{x^2} + 9{y^2} = 45$ is

An ellipse passes through the point $(-3, 1)$ and its eccentricity is $\sqrt {\frac{2}{5}} $. The equation of the ellipse is

Let $P(a\sec \theta ,\;b\tan \theta )$ and $Q(a\sec \varphi ,\;b\tan \varphi )$, where $\theta + \phi = \frac{\pi }{2}$, be two points on the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$, then $k$ is equal to

If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$ , the locus of $P (h, k)$ is a conic $C$ whose eccentricity 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 $…….$.