Find the locus of a point which moves such that its distance from the point $(0, 0)$ is twice its distance from the $y$-axis.

Find the equation of one of the common tangents to the parabola $y^2 = 4x$ and the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$.

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:

If $e_1$,$e_2$,and $e_3$ are eccentricities of the conics $y = x^2 - x + 3$,$\frac{x^2}{a^2} + \frac{y^2}{3a^4} = 1$,and $a^2x^2 - 3a^4y^2 = 1$ respectively,then which of the following is correct? (where $a > 1$)

$AB$ is a double ordinate of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that $\Delta AOB$ (where $O$ is the origin) is an equilateral triangle. Then the eccentricity $e$ of the hyperbola satisfies:

The equation of the common tangent to the circle $x^{2}+y^{2}=2$ and the parabola $y^{2}=8x$ is