PQ is a double ordinate of the parabola y^2 = | vedclass

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(A) Let the coordinates of $P$ be $(h, k)$ and $Q$ be $(h, -k)$. Since $P$ lies on the parabola $y^2 = 4ax$,we have $k^2 = 4ah$.Let $(x, y)$ be a poin

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$9y^2 = 4ax$

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(A) Let the coordinates of $P$ be $(h, k)$ and $Q$ be $(h, -k)$. Since $P$ lies on the parabola $y^2 = 4ax$,we have $k^2 = 4ah$.Let $(x, y)$ be a point of trisection of $PQ$. The points of trisection divide $PQ$ in the ratio $1:2$ or $2:1$.For the point dividing $PQ$ in ratio $1:2$,the $y$-coordinate is $y = \frac{1(-k) + 2(k)}{1+2} = \frac{k}{3}$,so $k = 3y$.The $x$-coordinate is $x = h$.Substituting $k = 3y$ and $h = x$ into the parabola equation $k^2 = 4ah$,we get $(3y)^2 = 4ax$,which simplifies to $9y^2 = 4ax$.

$PQ$ is a double ordinate of the parabola $y^2 = 4ax$. The locus of the points of trisection of $PQ$ is

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