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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 ? ( main issue - how to eliminate t to get the equation of locus )

vansh kharbanda , 10 Years ago

Grade Upto college level

1 Answers

Jitender Singh

Last Activity: 10 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

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