A normal to the hyperbola, $4x^2 – 9y^2\, = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively . If the parallelogram $OABP$ ( $O$ being the origin) is formed, then the locus of $P$ is

Let $e_{1}$ and $e_{2}$ be the eccentricities of the ellipse, $\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1(b<5)$ and the hyperbola $\frac{ x ^{2}}{16}-\frac{ y ^{2}}{ b ^{2}}=1$ respectively satisfying $e _{1} e _{2}=1 .$ If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha, \beta)$ is equal to

The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. The point of the tangency being $(a, b)$ then find the value of $\sin ^{-1}\left(\frac{a}{b}\right)$

The equation of the hyperbola whose foci are the foci of the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{9} = 1$ and the eccentricity is $2$, is

Let $H$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is