EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Ellipse>JEE Advanced Paper 1 - 2019>Q 4

HARD

JEE Advanced

IMPORTANT

Earn 100

Define the collections $\{E_{1}, E_{2}, E_{3}, . . . .\}$ of ellipses and $\{R_{1}, R_{2}, R_{3}, . . . .\}$ 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?

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Important Questions on Ellipse

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Ellipse>JEE Advanced Paper 1 - 2017>Q 5

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

, then which of the following options is the only Correct combination?