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Question

A normal chord AB of a parabola y212x=0 subtends a right angle at the vertex of the parabola. If the point of intersection of the normals drawn at A and B is (p,q), then the value of p2q2 is

A
2
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B
1
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C
12
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D
14
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Solution

The correct option is A 2

y2=12x ; y2=4axa=3A:(3t21,6t1), B:(3t22,6t2)

OA is perpendicular to OB
mOA×mOB=16t13t21×6t23t22=1t1t2=4 (1)

If AB is normal at A and intersects the parabola at B, then
t2=t12t1t1t2=t212t1=±2 [From (1)]t2=22

B is the point of intersection of the two normals at A and B
Coordinates of B is (24,122)=(p,q)
p2q2=2

If AB is normal at B and intersects the parabola at A, then
t1=t22t2t1t2=t212t2=±2t1=22

A is the point of intersection of the normals at A and B
Coordinates of A is (24,122)
p2q2=2



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