Define the collections $\left\{\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}, \ldots ..\right\}$ of ellipses and $\left\{\mathrm{R}_{1}, \mathrm{R}_{2}, \mathrm{R}_{3}, \ldots ..\right\}$ of rectangles as follows : $\mathrm{E}_{1}: \frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1$ ; $\mathrm{R}_{1}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $\mathrm{E}_{1}$; $\mathrm{E}_{\mathrm{n}}$ : ellipse $\frac{\mathrm{x}^{2}}{\mathrm{a}_{\mathrm{n}}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}_{\mathrm{n}}^{2}}=1$ of largest area inscribed in $\mathrm{R}_{\mathrm{n}-1}, \mathrm{n}>1$; $\mathrm{R}_{\mathrm{n}}$ : rectangle of largest area, with sides parallel to the axes, inscribed in $\mathrm{E}_{\mathrm{n}}, \mathrm{n}>1$. Then which of the following options is/are correct ?

(C) The length of latus rectum of $\mathrm{E}_{9}$ is $\frac{1}{6}$; (D) $\sum_{\mathrm{n}=1}^{\mathrm{N}}$ (area of $\mathrm{R}_{\mathrm{n}}$ ) < 24, for each positive integer N

(A) The eccentricities of $\mathrm{E}_{18}$ and $\mathrm{E}_{19}$ are NOT equal

(B) The distance of a focus from the centre in $\mathrm{E}_{9}$ is $\frac{\sqrt{5}}{32}$

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Ellipse

WhatsApp Email

Define the collections

{E_{1}, E_{2}, E_{3}, \dots ..}

of ellipses and

{R_{1}, R_{2}, R_{3}, \dots ..}

of rectangles as follows :

E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1

;

R_{1}

: rectangle of largest area, with sides parallel to the axes, inscribed in

E_{1}

;

E_{n}

: ellipse

\frac{x ^{2}}{a _{n}^{2}} + \frac{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 axes, inscribed in

E_{n}, n > 1

.
Then which of the following options is/are correct ?

(A)The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal
(B)The distance of a focus from the centre in $E_{9}$ is $\frac{5}{32}$
(C)The length of latus rectum of $E_{9}$ is $\frac{1}{6}$ ✓ Correct
(D) $\sum_{n = 1}^{N}$ (area of $R_{n}$ ) < 24, for each positive integer N✓ Correct

Answer: (C), (D)

Solution

For maximum area of rectangle inscribed in an ellipse, the vertices are at

(\pm a cos θ, \pm b sin θ)

where

θ = \frac{π}{4}

.

For

E_{1}

: vertices of

R_{1}

are at

(\pm \frac{3}{2}, \pm \frac{2}{2})

.

The pattern continues with

a_{n} = \frac{3}{( 2 ) ^{n - 1}}

and

b_{n} = \frac{2}{( 2 ) ^{n - 1}}

.

All ellipses have the same eccentricity

e = 1 - \frac{4}{9} = \frac{5}{3}

.

For

E_{9}

: distance of focus from center

= a_{9} e = \frac{3}{2 ^{4}} \cdot \frac{5}{3} = \frac{5}{16}

✗

Length of latus rectum

= \frac{2 b _{9}^{2}}{a _{9}} = \frac{2 \cdot \frac{4}{256}}{\frac{3}{16}} = \frac{1}{6}

✓

\sum_{n = 1}^{\infty}

Area of

R_{n} = 12 (1 + \frac{1}{2} + \frac{1}{4} + \dots) = 24

, so any finite sum is less than 24. ✓

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · One or more correct options

Practise more: all JEE Advanced PYQs on Ellipse →

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Ellipse

Solution

More JEE Advanced questions on Ellipse