The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin,respectively. If the point $(1, k)$ lies on the parabola,then a possible value of $k$ is :-

If $x = t^{2} + 2$ and $y = 2t$ represent the parametric equations of a parabola,find its Cartesian equation.

$A$ normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$,where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$,then the value of $24a$ is. . . .

The point $(3,4)$ is the focus and $2x - 3y + 5 = 0$ is the directrix of a parabola. Its latus rectum is:

The equation of the parabola with the focus $(3,0)$ and the directrix $x+3=0$ is:

If the tangents at the extremities of a chord $PQ$ of a parabola intersect at $T$,then the distances of the focus of the parabola from the points $P, T, Q$ are in