Paragraph: On the ellipse E= x^2 64 + y^2 9 =1 , tangents drawn at the point P₁, P₂, P₃, P_n on the ellipse intersecting the major axis at T₁, T₂, T₃ , T_n respectively. Question: If the value of _ i=1 ^n Area ( P_i T_i S ) Area ( P_i T_i S^ ) (P_i T_i )^2 =18 . where S and S^ represents the foci of the ellipse, then ' n ' equal to:

Step-by-step solution

aligned & (i) S A P_i T_i & S^ B P_i T_i aligned aligned & A₁= Area ( S P_i T_i )= 1 2 S A P_i T_i & A₂= Area ( S^ P_i T_i )= 1 2 S^ B P_i T_i & A₁ A₂ (P_i T_i )^2 = b^2 4 (S A S^ B=b^2 ) & A₁ A₂ (P_i T_i )^2 =18 n b^2 4 =18 n=8 aligned (ii) Tangent at (a e, b^2 a ) x a e a^2 + y b^2 a b^2 =1 aligned & e x+y=a & A= 2 a e 2 a 2 = 2 a^2 e =2 64 8 55 =64 & aligned