Let S and S ^ be the foci of the ellipse x^ 2 25 + y^ 2 9 =1 and P ( , ) be a point on the ellipse in the first quadrant. If ( SP )^ 2 + ( S ^ P )^ 2 - SP S ^ P =37, then ^ 2 + ^ 2 is equal to :

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Correct answer: D. 13

For the ellipse x^2 25 + y^2 9 = 1, we have a = 5, b = 3, c = 4. So foci are at S( 4, 0). For point P on the ellipse: SP + S'P = 10. Given (SP)^2 + (S'P)^2 - SP S'P = 37 Let r_1 = SP and r_2 = S'P. From (r_1 + r_2)^2 = 100, we get r_1^2 + r_2^2 = 100 - 2r_1r_2. Substituting into the given equation: 100 - 3r_1r_2 = 37, so r_1r_2 = 21. Thus r_1 = 3, r_2 = 7. From ( + 4)^2 + ^2 = 9 and ( - 4)^2 + ^2 = 49, subtracting gives 16 = -40, so = - 5 2 . From the ellipse equation: ^2 = 27 4 . Therefore ^2 + ^2 = 25 4 + 27 4 = 13.

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