The line $x - y + 2 = 0$ touches the parabola $y^2 = 8x$ at the point

The length of the latus rectum of the parabola $9x^2 - 6x + 36y + 19 = 0$ is:

If the focus of a parabola is $(0,-3)$ and its directrix is $y=3$,then its equation is

$A$ parabola has its focus on the positive $X$-axis and the $Y$-axis as its directrix. If $P(\alpha, 4)$ is a point on this parabola such that the tangent to the parabola at point $P$ passes through the origin, then the distance of $P$ from the origin is

Find the parameter $t$ for the point $(4, -6)$ on the parabola $y^2 = 9x$.

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 . . . . . .