From a point $(d, 0)$, three normals are drawn to the parabola $y^{2} = x$. Then:

If the vertex of a parabola is $(0, a)$ and the focus is $(0, 0)$,what is its equation?

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:

Find the equation of the parabola which is symmetric about the $y$-axis and passes through the point $(2, -3)$.

Find the vertex,focus,axis,length of latus rectum,and the equation of the directrix for the parabola $y^{2} = 4x + 4y$.

If $2x + y + \lambda = 0$ is a focal chord of the parabola $y^2 = -8x$,find the value of $\lambda$.