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For the ellipse x²/25 + y²/9 = 1 (centre C), let P be the point with eccentric angle pi/4. The normal at P meets the major and minor axes at A and B; N1, N2 are feet of perpendiculars from foci S1, S2 onto the tangent at P; N is the foot of perpendicular from C onto the normal at P; the tangent at P meets the x-axis at T. Match List-I to List-II. List-I: (I) (CA)(CT) (II) (PN)(PB) (III) (S1N1)(S2N2) (IV) (S1P)(S2P) List-II: (P) a² = 25 (Q) e² a² = 16 (R) b² = 9 (S) other (T) other Which matching is correct?

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

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

Solution

By known ellipse properties (S1N1)(S2N2) = b² = 9, and (CA)(CT) = a² e² = 16; matching the remaining products to (P) and (S) gives the first option.

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