Question

Let $a,r,s$ and $t$ be non-zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ and $S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is point $(2a,0)$.
If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

Answer 1

Equation of tangent and normal at $(a{t}^2,2at)$are given by $ty=x+a{t}^2$ and $y+tx=2at+a{t}^3,$ respectively.
Tangent at $P:ty=x+a{t}^2$ or $y=\frac{x}{t}+at$
Normal at $S:y+\frac{x}{t}=\frac{2a}{t}+\frac{a}{t^3}$
Solving $2y=at+\frac{2a}{t}+\frac{a}{t^3}$
$\Rightarrow y=\frac{a(t^2+1{)}^2}{2t^3}$