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Question

From the vertex of the parabola y2=4ax pair of perpendicular chords are drawn. If this chords are adjacent sides of a rectangle, then the locus of the vertex of the rectangle diagonally opposite to the vertex of parabola is

A
y2=4a(x4a)
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B
y2=4a(x6a)
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C
y2=4a(x2a)
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D
y2=4a(x8a)
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Solution

The correct option is D y2=4a(x8a)
Given parabola is y2=4ax
Let the vertex be O=(0,0) and the chords be OA and OB
Where A=(at21,2at1),B=(at22,2at2)

Now, the slope of the line OA is
m1=2t1
Slope of the line OB is
m2=2t2 m1.m2=1 ( OAOB)t1t2=4

Let the coordinates of the vertex opposite to the vertex of parabola be C=(h,k)
As the diagonals of a rectangle bisect each other, so
(h2,k2)=(a(t21+t22)2,2a(t1+t2)2)ha=t21+16t21, k2a=t14t1(k2a)2=ha8

Hence, the locus is
y2=4a(x8a)

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