JEE Advanced 2019 · Paper 1 · Mathematics Q9

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MathematicsMultiple correct+4 / -1Question 45 of 54

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:

$E_1: \dfrac{x^2}9 + \dfrac{y^2}4 = 1$ ;

$R_1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$ ;

$E_n$ : ellipse $\dfrac{x^2}{a_n^2} + \dfrac{y^2}{b_n^2} = 1$ 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?

The eccentricities of

E_{18}

and

E_{19}

are NOT equal

\displaystyle\sum_{n=1}^N(\text{area of }R_n) < 24

, for each positive integer

N

The length of latus rectum of

E_9

\dfrac16

The distance of a focus from the centre in

E_9

\dfrac{\sqrt5}{32}

Answer and worked solution

Official answer: B, C

The pattern. The largest rectangle in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ has corners $\left(\frac a{\sqrt2},\frac b{\sqrt2}\right)$ and area $2ab$ . The largest ellipse inside that rectangle has semi-axes $\frac a{\sqrt2}$ and $\frac b{\sqrt2}$ . So
$a_n=\frac3{2^{(n-1)/2}},\qquad b_n=\frac2{2^{(n-1)/2}}.$

The options.

(A) The ratio $b_n/a_n$ is fixed, so every $E_n$ has the same eccentricity. False.
(B) Area of $R_n=2a_nb_n=\dfrac{12}{2^{n-1}}$ . The partial sums are below $\frac{12}{1-\frac12}=24$ . True.
(C) $a_9=\frac3{16}$ and $b_9=\frac18$ , so the latus rectum is $\dfrac{2b_9^2}{a_9}=\dfrac{2/64}{3/16}=\dfrac16$ . True.
(D) The focal distance is $\sqrt{a_9^2-b_9^2}=\dfrac{\sqrt5}{16}$ , not $\dfrac{\sqrt5}{32}$ . False.

(B), (C)

Topics:Ellipse Geometric progression Maxima and minima

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JEE Advanced 2019 · Paper 1, Mathematics question 9