What is the length of the chord of contact of the tangents drawn from the point $(-2, -1)$ to the parabola $y^2 = 4x$?

If two normals to the parabola $y^2 = 4x$ passing through the point $(15, 12)$ are $4x + y = 72$ and $3x - y = 33$,find the third normal.

Let $A(1, 2)$ be a point on the parabola $y^2 = 4x$. Let $B$ and $C$ be the points of intersection of this parabola with a variable line passing through the point $P(5, -2)$. Then the $\Delta ABC$ (if it exists):

If $a \neq 0$ and the line $2bx + 3cy + 4d = 0$ passes through the points of intersection of the parabolas $y^2 = 4ax$ and $x^2 = 4ay$,then

The length of the latus rectum of the parabola $y^2+8x-2y+17=0$ is

The equation of the parabola with focus $(0,0)$ and directrix $x+y=4$ is