Question

PQ is a double ordinate of a parabola. Find the locus of its point of trisection.

Answer 1

Let $P(h,k)$ be the point which trisect the double ordinate $L{L}^{\prime }$

Coordinates of $L$ are $(a{t}^2,2at)$ and $L^{\prime }$ are $(a{t}^2,-2at)$

Coordinates of point which divides $L{L}^{\prime }$ in $1:2$ are

$P(\frac{a{t}^2\left(1\right)+a{t}^2\left(2\right)}{2+1},\frac{-2at\left(1\right)+2at\left(2\right)}{2+1})\Rightarrow P(a{t}^2,\frac{2at}{3})h=a{t}^2......\left(i\right)k=\frac{2at}{3}\Rightarrow t=\frac{3k}{2a}$

Substituting $t$ in $\left(i\right)$

$h=a{\left(\frac{3k}{2a}\right)}^{2}h=a\frac{9k^2}{4a^2}\Rightarrow 9k^2=4ah$

Replacing $h$ by $x$ and $k$ by $y$

$9y^2=4ax$

is the required locus