For some $\theta \in\left(0, \frac{\pi}{2}\right),$ if the eccentricity of the hyperbola, $x^{2}-y^{2} \sec ^{2} \theta=10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^{2} \sec ^{2} \theta+y^{2}=5,$ then the length of the latus rectum of the ellipse is

Let the ellipse $E : x ^2+9 y ^2=9$ intersect the positive $x$ - and $y$-axes at the points $A$ and $B$ respectively Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle which vertices $A, P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to

The length of the axes of the conic $9{x^2} + 4{y^2} - 6x + 4y + 1 = 0$, are

If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{4 a^{2}}=1$ and the co-ordinate axis is $kab,$ then $\mathrm{k}$ is equal to ..... .

An ellipse having foci at $(3, 3) $ and $(- 4, 4)$  and passing through the origin has eccentricity equal to

The locus of a variable point whose distance from $(-2, 0)$ is $\frac{2}{3}$ times its distance from the line $x = - \frac{9}{2}$, is