What is the equation of the common tangent to the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $X$-axis?

Let the eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be the reciprocal of the eccentricity of the hyperbola $2x^2 - 2y^2 = 1$. If the ellipse intersects the hyperbola at right angles,then the square of the length of the latus-rectum of the ellipse 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:

The quadratic equation whose roots are $l$ and $m$,where $l = \lim_{\theta \rightarrow 0} \left( \frac{3 \sin \theta - 4 \sin^2 \theta}{\theta} \right)$ and $m = \lim_{\theta \rightarrow 0} \frac{2 \tan \theta}{\theta(1 - \tan^2 \theta)}$,is:

Find the angle of intersection of the curves $y^{2}=4ax$ and $x^{2}=4by$.

Let $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b$ and $H : \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$. Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a - A = 2$,and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$,then the sum of the lengths of their latus rectums is equal to :