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Question

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2=4ax is another parabola whose directrix is x=k. The value of k is

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Solution

Focus of the parabola y2=4ax is S(a,0)
And point P on the parabola is taken as (at2,2at)
Now, mid point of SP is (a+at22,at)=R(h,k), say
a(1+t2)=2h and k=att=k/a
Now eliminating t we get, a(1+k2/a2)=2h
k2=2aha2
Thus locus of R is y2=2axa2
y2=2a(xa/2)
Thus directrix of this parabola is, xa/2=(a/2)x=0

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