What is the length of the latus rectum of the parabola $x^2 = -y$?

The line $x - y + 2 = 0$ touches the parabola $y^2 = 8x$ at the point

What is the length of the chord intercepted by the parabola $y = x^2 + 3x$ on the line $x + y = 5$?

If ${y_1}$ and ${y_2}$ are the ordinates of two points $P$ and $Q$ on the parabola ${y^2 = 4ax}$ and ${y_3}$ is the ordinate of the point of intersection of the tangents at $P$ and $Q$,then:

If $a > 0$ and $b^2 - 4ac = 0$,then the curve $y = ax^2 + bx + c$

The equation of the line touching both parabolas $y^2=4x$ and $x^2=-32y$ is