the locus of the mid point of the line segment joining the focus to a moving point on the parabola y^2 = 4ax is another parabola with directrix ? ( main issue - how to eliminate t to get the equation of locus )

Jitender Singh IIT Delhi
8 years ago
Ans:
Focus O of the parabola:
$(a, 0)$
Moving point P on the parabola:
$(at^{2}, 2at)$
Let the mid point of OP to be:
$(h, k)$
$h = \frac{at^{2}+a}{2}, k = at$
$2h = at^{2} + a$
$t = \frac{k}{a}$
$k^{2} = 2ah - a^{2}$
Replace
$h \rightarrow x, k \rightarrow y$
$y^{2} = 2ax - a^{2}$
$x = 0$
is the directrix.
Thanks & Regards
Jitender Singh
IIT Delhi