The locus of the midpoint of the line segment joining the focus of the moving point on the parabola y

2

=4ax is another parabola with directrix :

Let midpoint (h,k)

Let moving point be (at

2

,2at)

And let focus (a,0)

h=

2

at

2

+a

k=

2

2at+0

=at

h=

2

a

[

h

2

k

2

+1]

2ah−a

2

=k

2

y

2

=2a(x−

2

a

)

y

2

=2ax

Directrix is :

x−

2

a

=−

2

a

x=0

Hence x=0 is another directrix of other parabola.

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