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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

Arun
25763 Points
2 years ago

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.

Regards
Arun