The point of contact of the tangent to the parabola $y^2 = 4ax$ which makes an angle of $60^\circ$ with the $x$-axis is:

Three normals drawn from any point to the parabola $y^2 = 4ax$ cut the line $x = 2a$ in points whose ordinates are in arithmetical progression. Then the tangents of the angles which the normals make with the axis of the parabola are in:

If $y=4x+3$ is parallel to a tangent to the parabola $y^{2}=12x$,then its distance from the normal parallel to the given line is

Let $y^2=12x$ be a parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of circle $C$ is $64x^2+64y^2-\alpha x-64\sqrt{3}y=\beta$,then $\beta-\alpha$ is equal to . . . . . .

The equation of the parabola with vertex at $(0,0)$ and length of latus rectum equal to $\frac{16}{3}$ is:

The vertex of a parabola is the point $(a, b)$ and the latus rectum is of length $l$. If the axis of the parabola is along the positive direction of the $y$-axis,then its equation is