If the midpoint of a chord is $(-1, 1)$,find the equation of the chord of the parabola $y^2 = 6x$.

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

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If there are two more normals to this parabola which pass through $(5,0)$,the centroid of the triangle formed by the feet of these three normals is

The length of the latus rectum of the parabola ${y^2} - 4y - 2x - 8 = 0$ is

If the equation of a system of parallel chords of the parabola $y^2 = \frac{2}{3}x$ is $y + 2x + 1 = 0$,find its diameter.