The vertex of the parabola $3x - 2y^2 - 4y + 7 = 0$ is

The line $x-2y-3=0$ cuts the parabola $y^2=4ax$ at the points $P$ and $Q$. If the focus of this parabola is $(\frac{1}{4}, k)$,then $PQ=$

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 focus $(a, b)$ and directrix $\frac{x}{a} + \frac{y}{b} = 1$ is given by

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2$,then the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

If the line $y=2x+k$ is a normal to the parabola $y^2=4x$,then $k=$