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Question

The locus of the midpoint of the focal distance of a variable point moving on the parabola y2=4ax is a parabola whose

A
Latus rectum is half the latus rectum of the original parabola
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B
Vertex is (a2,0)
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C
Directrix is yaxis
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D
Focus has the co-ordinates (a,0)
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Solution

The correct option is D Focus has the co-ordinates (a,0)
Any point on the parabola is
P(at2,2at)
Therefore, the midpoint of S(a,0) and P(at2,2at) is,
R(a+at22,at)(h,k)
h=a+at22,k=at
Eliminating t,
2x=a(1+y2a2)=a+y2a
2ax=a2+y2
y2=2a(xa2)
It is a parabola with vertex at (a2,0) and latus rectum 2a.
The directrix is
xa2=a2
x=0
The focus is
xa2=a2
x=a
Therefore the focus is (a,0)

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