At which point does the line $x + y = 1$ intersect the parabola $y = x - x^2$?

Let $x+y=k$ be a normal to the parabola $y^2=12x$. If $p$ is the length of the perpendicular from the focus of the parabola onto this normal,then $4k-2p^2$ is equal to

The point on the curve $4y^2 - 4y + 2x - 1 = 0$ at which the tangent becomes parallel to the $Y$-axis is:

If $lx + my + n = 0$ is tangent to the parabola $x^2 = y$,then the condition of tangency is

The equation of the tangent at $P(-4, -4)$ on the curve $x^{2} = -4y$ is

Let $O$ be the vertex of the parabola $y^2 = 4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is a conic $C$, then the length of its latus rectum is: