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$\mathrm{P Q}$ is a double ordinate of a parabola $\mathrm{ y^2=4 a x}$ . Then, the locus of its point of trisection is
Option: 1
$\mathrm{y^2=4 a x}$

Option: 2
$\mathrm{9 y^2=4 a x}$

Option: 3
$\mathrm{3 y^2=4 a x}$

Option: 4
$\mathrm{3 y^2=a x}$

Answers (1)

best_answer

Let $\mathrm{A(h, k)}$ be the point of trisection of the double ordinate $\mathrm{P Q.}$ Then, the point of intersection with the axis of parabola is $\mathrm{L(h, 0)}$

$\mathrm{. \begin{aligned} \Rightarrow \quad A L & =k \\ & =B L \\ \therefore \quad A B & =A L+B L=k+k=2 k \end{aligned} }$

As A and B are the points of trisection,

$\mathrm{ \begin{array}{lll} \therefore & A P & =A B=B Q=2 k \\\\ \therefore & L P & =L A+A P=k+2 k=3 k \end{array} }$

$\mathrm{\Rightarrow \quad P \text { is }(h, 3 k) \text { which lies on } y^2=4 a x}$

The condition is $\mathrm{9 k^2=4 a h}$

$\mathrm{\therefore \quad}$ the locus of $\mathrm{A(h, k)}$ is

$\mathrm{ 9 y^2=4 a x }$

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