Question

Let $PQ$ be the chord of the parabola $y ^2=4 x$ with slope $=2$. Locus of the point $R$ which divides the chord $PQ$ intemally in the ratio $1: 2$ is a parabola whose length of latus rectum is $L$. Find the value of $18 L$.

Answer 1

$ \text { Given } \frac{2}{ t _1+ t _2}=2 $
$\therefore t _1+ t _2=1$ ....(1)
$\text { Also } 3 h = t _1^2+2 t _2^2 $ ....(2)
$3 k =2 t _1+4 t _2 $
$3 k =2\left( t _1+2 t _2\right)$ .....(3)
$\text { Now, } 3 k =2\left(1+ t _2\right) $
$\frac{3 k }{2}-1= t _2$
$\therefore t _1=1- t _2=2-\frac{3 k }{2} $
$3 h = t _1^2+2 t _2^2$
$3 h =\left(2-\frac{3 k }{2}\right)^2+2\left(\frac{3 k }{2}-1\right)^2$
$3 h =4+\frac{9 k ^2}{4}-6 k +\frac{9 k ^2}{2}+2-6 k $
$3 h =\frac{27 k ^2}{4}-12 k +6 $
$3( x -2)=\frac{27}{4}\left( y ^2-\frac{16 y }{9}\right) $
$( x -2)=\frac{9}{4}\left[\left( y -\frac{8}{9}\right)^2-\frac{64}{81}\right] $
$\frac{4}{9}( x -2)=\left( y -\frac{8}{9}\right)^2-\frac{64}{81}$
$\therefore \left(y-\frac{8}{9}\right)^2=\frac{4}{9}\left(x-\frac{2}{9}\right)$
hence length of latus recturm $=\frac{4}{9}= L$
$\Rightarrow 18 L =8$.