The line $lx + my + n = 0$ will be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

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 tangency is $(a, b)$. Then the eccentricity of the conic is:

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$,where $l$ and $m$ are co-prime numbers,then $l^2+m^2+n^2$ is equal to . . . . . . .

The distance of the focus of $x^{2}-y^{2}=4$ from the directrix which is nearer to it,is

The number of normals that can be drawn from an external point to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is:

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$,where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola,in the first quadrant. Let $\angle SPS_1 = \alpha$,with $\alpha < \frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola,intersects the straight line $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$,and $\beta = S_1P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is: