Hyperbola

If the normal to the rectangular hyperbola \(x^2-y^2=1\) at the point \(P(\pi / 4)\) meets the curve again at \(Q(\theta)\), then \(\sec ^2 \theta+\tan \theta=\)

If the vertices and foci of a hyperbola are respectively \(( \pm 3,0)\) and \(( \pm 4,0)\), then the parametric equations of that hyperbola are

If one focus of a hyperbola is \((3,0)\), the equation of its directrix is \(4 x-3 y-3=0\) and its eccentricity \(e=5 / 4\), then the coordinates of its vertex is

If the focal chord of the hyperbola subtends a right angle at the center, then its eccentricity is

The asymptotes of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), with any tangent to the hyperbola form a triangle whose area is \(a^2 \tan (\alpha)\). Then, its eccentricity equals