If $(0, 4)$ and $(0, 2)$ are the vertex and focus of a parabola respectively,then what is its equation?

If the area of the triangle whose one vertex is at the vertex of the parabola,${y^2} + 4(x - {a^2}) = 0$ and the other two vertices are the points of intersection of the parabola and $y$-axis,is $250 \text{ sq. units}$,then a value of $a$ is

If $x+y=k$ is a normal to the parabola $y^{2}=12x$,then the value of $k$ is:

Statement $I$: $4x^2+y^2-4xy-30x-50y+40=0$ is the equation of a parabola having $(2,3)$ as its focus and $x+2y+5=0$ as its directrix.
Statement $II$: The equation of the directrix of the parabola $x^2-4x+16y+52=0$ is $y+1=0$.
Which of the above statements is (are) true?

The line $x \cos \alpha + y \sin \alpha = p$ will touch the parabola $y^2 = 4a(x + a)$ if:

Find the equation of the normal to the parabola $y^2 = 8x$ at the point $(2, 4)$.