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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 with directrix

A
x=a
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B
x=a2
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C
x=0
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D
x=a2
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Solution

The correct option is C x=0
Let P(h,k) be the mid-point of the line segment joining the focus (a,0) and a general point Q(x,y) on the parabola. Then,
h=x+a2,k=y2x=2ha,y=2k.
Put these values of x and y in y2=4ax, we get
4k2=4a(2ha)
4k2=8ah4q2k2=2aha2
So, locus of P(h,k) is y2=2axa2
y2=2a(xa2)
Its directrix is xa2=a2x=0

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