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Question

The locus of the midpoint of the line joining focus and any point on the parabola y2=4ax is a parabola with the equation of directrix as


A

x+a=0

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B

2x+a=0

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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


Explanation of the correct option.

Compute the directrix:

Let the point P(h,k) be the midpoint of the line segment joining the focus (a,0)and a general point Q(x,y) on the parabola.

Therefore,

h=[x+a]2x=2h-a

k=y2y=2k

Substitute the values of x and y in y2=4ax,

4k2=4a(2ha)4k2=8ah-4a2k2=2ah-a2

The locus of P(h,k)is

y2=2axa2.

y2=2ax-a2

Therefore, Its directrix is

x-a2=a2x=0.

Hence option C is the correct answer.


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