Question

$PQ$ is a double ordinates of a parabola. Find the locus of point trisecting it.

Answer 1

Given, equation of parabola
$y^2=4ax$
Let $R,S$ trisects double ordinate $PQ$.
Let double ordinate $PQ$ meets $A-$ axis at point $A$.
Let coordinates of point $R$ are $(h,k)$
then $OA-h,AR=k$
$\therefore RS=RA+AS=k+k-2k$
$\Rightarrow PR=RS=SQ=2k$
$\Rightarrow PA=RA+PR=k+2k=3k$
$\therefore$ Coordinates of point $P$ are $(h,3k)$,
$\therefore$ Point $P$ lies on parabola
$\therefore (3k{)}^2=4ah\Rightarrow 9k^2=4ah$
Thus, required locus is $9y^2=4ax$.