Question

# A variable chord PQ of the parabola y2=4ax is drawn parallel to line y=x. Then the locus of point of intersection of normals at P and Q is:

A
2xy12a=0
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B
2x+ya=0
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C
x2y4a=0
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D
3xy8a=0
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Solution

## The correct option is A 2x−y−12a=0Equation of normal's at P(at21,2at2) and Q(at22,2at2) are y+xt1=2at1+at31 and y+xt2=2at2+at32 Point of intersection of both lines is x=2a+a(t21+t22+t1t2) and y=−at1t2(t1+t2) ∵ Slope of chord PQ=2at1−2at2at21−at22=1 ∴ t1+t2=2 ⇒ y=−at1t2(t1+t2)⇒ t1t2=−y2a x=2a+a(t21+t22+t1t2)⇒ x=2a+a((t1+t2)2−t1t2) ⇒ x=2a+a(4+y2a) 2x−y−12a=0

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