Parametric Equation of conics: Parabola

Question

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$.

a) Show that $pq=-1$

b) If $P$ is joined to the vertex, $V$, and $PV$ is produced to meet the directrix at $D$, show that $DQ$ is parallel to the axis of the parabola.

Answer 1

I wonder if the following hints help you?

Hint 1. What is the gradient of the line $PQ$?, to begin with

\begin{align} \frac{ap^{2}−aq^{2}}{2ap−2aq} &= \frac{a(p^{2}−q^{2})}{2a(p−q)} \\ &=\quad... \end{align}

Hint2 What, therefore is the equation of the straight line of the chord?

Hint 3 The given chord will be a focal chord if (say, the point) $(0,a)$ lies on it.

Answer 2

For part a), if segment $PQ$ contains the focus $F=(a,0)$ then: $$ y_P:(x_P-a)=y_Q:(x_Q-a), \quad\hbox{whence}\quad pq(p-q)=q-p. $$ For part b) one has $D=(-a,y_D)$, where $$ y_D:x_D=y_P:x_P, \quad\hbox{so that}\quad y_D=-{2a\over p}=2aq=y_Q. $$