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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

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
 

Grade:11

1 Answers

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

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