Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$; focus $(3, 0)$.

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 . . . . . .

$A$ line passes through the point $(-1, 1)$ and makes an angle $\sin^{-1}(\frac{3}{5})$ in the positive direction of the $x$-axis. If this line meets the curve $x^2 = 4y - 9$ at points $A$ and $B$,then the length $|AB|$ is equal to

For what value of $k$ does the line $2y - x + k = 0$ touch the parabola $x^2 + 4y = 0$?

If the focus and directrix of a parabola are $(3, 5)$ and $x + y = 4$,then the coordinates of its vertex are

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $x^{2}=-9y$.