If the latus rectum through one of the foci of a hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtends a right angle at the farther vertex of the hyperbola, then $b^2=$

Let $P, Q, R, S$ be the points of intersection of the circle $x^2+y^2=4$ and the hyperbola $x y=\sqrt{3}$. If $P=(\alpha, \beta)$ and $\alpha>\beta>0$, then the equation of the tangent drawn at $P$ to the hyperbola is

If the tangent drawn at the point $P(3 \sqrt{2}, 4)$ on the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$ meets its directrix at $Q(\alpha, \beta)$ in fourth quadrant, then $\beta=$

If $l$ is the maximum value of $-3 x^2+4 x+1$ and $m$ is the minimum value of $3 x^2+4 x+1$, then the equation of the hyperbola having foci at $(l, 0),(7 m, 0)$ and eccentricity as 2 is