Algebraically proving distances on origin-centered parabola.

Question

$P$ is the point $(2ap,ap^2)$ on the parabola $x^2=4ay$. The tangent at $P$ meets the $y$ axis at $T$ and $PN$ is drawn perpendicular to the $y$ axis, meeting it at $N$. The directrix meets the $y$ axis at $A$.

Prove:

$(i)$: $OS=ON$

$(ii)$: $ON=OT$

$(iii)$: The $x$ axis bisects $PT$ at a point, say $B$.

Answer 1

let the equation of tangent at point $P$ be $y=px-ap^2$ . it intersects y axis at ($0,at$). coordinates of N will be ($0,ap^2$) coordinates of $A$ will be ($0,-a$)

Find the distances through distance formula and you will be able to prove it.