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For the parabola y² = 12x, match each item in List-I with the correct point in List-II. List-I: (I) The midpoint of T and G, where the tangent and normal at an end of the latus rectum meet the x-axis at T and G respectively; (II) The fixed point K on the axis such that for variable chords through K, the sum of the reciprocals of the squares of the two segments is constant; (III) The fixed point through which all chords subtending a right angle at the origin pass; (IV) The point through which the radical axis of the circles on AB and CD (chords meeting at a point E on the axis) as diameters always passes. List-II: (P) (0,0); (Q) (3,0); (R) (6,0); (S) (12,0).

1. I -> Q; II -> R; III -> S; IV -> P
2. I -> P; II -> Q; III -> R; IV -> S
3. I -> S; II -> Q; III -> R; IV -> P
4. I -> R; II -> Q; III -> P; IV -> S

Correct answer: I -> Q; II -> R; III -> S; IV -> P

Solution

With a = 3: midpoint of T(-3,0) and G(9,0) is (3,0)=Q; the special K is (6,0)=R; chords subtending a right angle at the origin pass through (4a,0)=(12,0)=S; the radical axis passes through the vertex (0,0)=P.

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