$A$ point $P(x, y)$ moves such that its distance from the point $(a, 0)$ is always equal to its distance from the line $x + a = 0$. The locus of the point 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?

If the lines $y = x$ and $y = -x$ intersect the parabola $y^2 = 4x$ at points $A$ and $B$ respectively,other than the origin,what is the length of $AB$?

The equation of the tangent to the parabola $y^2 = 4ax$ at the point $(a/t^2, 2a/t)$ is:

The equation of the directrix of the parabola $y^2+4y+4x+2=0$ is

The length of the chord of the parabola $y^2 = x$ which is bisected at the point $(2, 1)$ is