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

The tangents to the curve $y = (x - 2)^2 - 1$ at its points of intersection with the line $x - y = 3$ intersect at the point:

Find the point of intersection of the parabola $y^2 = 2px$ and a circle whose center is at the focus of the parabola and which touches the directrix of the parabola.

If the lines $2x + 3y + 12 = 0$ and $x - y + k = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $k$ is equal to

If the straight line $x + y = 1$ touches the parabola $y^2 - y + x = 0$,then the coordinates of the point of contact are

If $PSQ$ is a focal chord of the parabola $y^2 = 8x$ such that $SP = 6$,then find the length of $SQ$.