Equation of ellipse `E_(1)` is `(x^(2))/(9)+(y^(2))/(4)=1`, A rectangle `R_(1)`, whose sides are parallel to co-ordinate axes is inscribed in `E_(1)` such that its area is maximum now `E_(n)` is an ellipse inside `R_(n-1)` such that its axes is along co-ordinate axes and has maxmim possible area `AA n ge 2, n in N`, further `R_(n)` is a rectangle whose sides are parallel to co-ordinate axes and is inscribed in `E_(n-1)`. Having maximum area `AA n ge 2, n in N`

Options

A. `underset(n=1)overset(m)sum` area of rectangle `(R_(n)) lt 24 AA m in N`
B. Length of latus rectum of `E_(9)=(1)/(6)`
C. Distance between focus and centre of `E_(9)=(sqrt(5))/(32)`
D. The eccentricities of `E_(18)` and `E_(19)` are not equal.

Answer

A, B

Explanation

Area max when `theta=45^(@)`
`{:(,a,b),(E_(1),3,2),(E_(2),(3)/(sqrt(2)),(2)/(sqrt(2))),(E_(3),(3)/((sqrt(2))^(2)),(2)/((sqrt(2))^(2))),(.,.,.),(E_(9),(3)sqrt((sqrt(2))^(8)),(2)/((sqrt(2))^(8))):}`
(A). `E_(1)+E_(2)+...E_(m)`
when `mtoinfty(2ab)/(1-(1)/(sqrt(2))(1)/(sqrt(2)))=4ab=4,3,2=24`
(B). Length of LR is ellipse `=(2b^(2))/(a)=2.(4.2^(4))/(2^(8).3)=(1)/(6)`
(C). Distance between focus and center of ellipse `=a_(9)e_(9)=(3)/(2^(4)).(sqrt(5))/(3)=(sqrt(5))/(16)`