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Question:

Define the collections E1, E2, E3,... of ellipses and R1, R2, R3,... of rectangles as follows: E1: x²/9 + y²/4 = 1; R1: rectangle of largest area with sides parallel to the axes, inscribed in E1; En: ellipse x²/an² + y²/bn² = 1 of largest area inscribed in Rn-1, n > 1; Rn: rectangle of largest area, with sides parallel to the axes, inscribed in En, n > 1. Then which of the following options is/are correct?

The eccentricities of E18 and E19 are NOT equal

The length of latus rectum of E9 is 16

∑n=1N(area of Rn) < 24, for each positive integer N

The distance of a focus from the centre in E9 is 5√32

Solution:

Correct option is C.∑n=N(area ofRn)<24, for each positive integerNArea Max whenθ=45∘abE132E23222E33(2)22(2)2: : :E93(2)82(2)8(A)E1+E2+.+whenm→∞2ab1−1;212=4ab=4.3.2=24(B) Length ofLRis ellipse=2b2a=2.4.2428.3=16(C) distance between focus and center of ellipse=a9e9=324.53=516.