Define the collections E 1 ,   E 2 ,   E 3 , . . . . of ellipses and R 1 ,   R 2 ,   R 3 , . . . . of rectangles as follows: E 1 : x 2 9 + y 2 4 = 1 ; R 1 : rectangle of largest area, with sides parallel to the axes, inscribed in E 1 ; E n : ellipse x 2 a n 2 + y 2 b n 2 = 1 of largest area inscribed in R n - 1 ,   n > 1 ; R n : rectangle of largest area, with sides parallel to the a

Step-by-step solution

As given: E 1 = x 2 a 2 + y 2 b 2 = 1 (Here a = 3 and b = 2 ) Semi major axis = a and semi minor axis = b Let a vertex of R 1 be a c o s θ , b s i n θ Then area of R 1 = 2 a c o s θ × 2 b s i n θ R 1 = 2 a b s i n 2 θ Maximum area of R 1 = 2 a b , when s i n 2 θ = 1 or, 2 θ = π 2 θ = π 4 Maximum area of R 1 = 2 a b ....(i) Now, ellipse E 2 will have semi major axis a 2 and semi minor axis b 2 ∵ E 2 = x 2 a 2 2 + y 2 b 2 2 = 1 , hence maximum area of R 2 = 2 a 2 b 2 Similarly, ellipse E 3 will have semi major axis a