Question

A fixed parabola $y^2=4 x$ touches a variable parabola. If the equation to the locus of the vertex of the variable parabola is another parabola $P$, then find the length of latus rectum of $P$.
Note: Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the $x$-axis.

Answer 1

Let the equation of the parabola with vertex $( h , k )$
$( y - k )^2=-l( x - h )$

$\text { substituting } x =\frac{ y ^2}{l} ; l=4 a $
$( y - k )^2=-l\left(\frac{ y ^2}{l}- h \right)$
$y ^2-2 ky + k ^2=- y ^2+l h $
$2 y ^2-2 ky + k ^2-l h =0 $
$\text { put } D =0 $
$4 k ^2-8\left( k ^2-l h \right)^2=0 $
$4 k ^2=8 l x \Rightarrow y ^2=2 l x =8 ax =8 x \Rightarrow \text { latus rectum=8 }$