# Parametric Equation of conics: Parabola

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.

• Write the equation of PQ and use that the focus lies on PQ. – SchrodingersCat Nov 10 '15 at 9:21

Hint 1. What is the gradient of the line $PQ$?, to begin with
Hint 3 The given chord will be a focal chord if (say, the point) $(0,a)$ lies on it.
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 $$y_D:x_D=y_P:x_P, \quad\hbox{so that}\quad y_D=-{2a\over p}=2aq=y_Q.$$