If the tangent at the point $(1,2)$ on the ellipse $3x^2+4y^2=19$ is also a tangent to the parabola $y^2-kx=0$,then $k=$

If a normal drawn to the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ touches the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$,then the square of the slope of that normal is

For some $\theta \in (0, \frac{\pi}{2})$, let the eccentricity and the length of the latus rectum of the hyperbola $x^{2} - y^{2} \sec^{2} \theta = 8$ be $e_{1}$ and $l_{1}$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^{2} \sec^{2} \theta + y^{2} = 6$ be $e_{2}$ and $l_{2}$, respectively. If $e_{1}^{2} = e_{2}^{2}(\sec^{2} \theta + 1)$, then $(\frac{l_{1}l_{2}}{e_{1}e_{2}}) \tan^{2} \theta$ is equal to . . . . . . .

The foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide. Then,the value of $b^2$ is

If the normal to a parabola $y^2 = 4ax$ at point $P$ meets the curve again at point $Q$,and if $PQ$ and the normal at $Q$ make angles $\alpha$ and $\beta$ respectively with the $x$-axis,then the value of $\tan \alpha (\tan \alpha + \tan \beta)$ is:

If a tangent to the hyperbola $xy = -1$ is also a tangent to the parabola $y^2 = 8x$,then the equation of that tangent is