Find the equation of the locus of R if the chord $PQ$ passes through $(0,a)$ The parabola is $x^2=4ay$
Information given: 
Points  of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola $x^2=4ay$. 
The equation of the tangent at P is $y=px-ap^2$
The equation of the tangent at Q is $y=qx-aq^2$
The equation of the normal at P is $x+py=2ap+ap^3$
The equation of the normal at Q is $x+qy=2aq+aq^3$
Normals of P and Q intersect at point R whose coordinates are $(-apq[p+q], a[p^2+pq+q^2+2])$
Find the equation of the locus of  R if the chord PQ passes through $(0,a)$

The line passing through $P$ and $Q$ is 
$$y-aq^2=\frac{aq^2-ap^2}{2aq-2ap}(x-2aq),$$
i.e.
$$y=\frac{p+q}{2}x-apq.$$
If $PQ$ passes through $(0,a)$, then we have
$$a=\frac{p+q}{2}\cdot 0-apq\quad\Rightarrow \quad pq=-1.$$
Here note that
$$R_x=-apq(p+q)=a(p+q)\quad\Rightarrow \quad p+q=\frac{R_x}{a}$$
and that
$$R_y=a(p^2+pq+q^2+2)=a((p+q)^2-pq+2)=a((p+q)^2+3)$$
