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

Find the point of contact of the line $2x - y + 2 = 0$ with the parabola $y^2 = 16x$.

If the directrix of the parabola $x^2+4y-6x+\lambda=0$ is $y+1=0$,then which of the following is correct?

The point on the axis of the parabola $3y^2+4y-6x+8=0$ from which $3$ real normals can be drawn is given by:

Let chord $PQ$ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$. If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin \alpha$ is equal to:

If $5x - 2y + k = 0$ is a tangent to the parabola $y^2 = 6x$,then their point of contact is