Define the collections {E₁, E₂, E₃, ...} of ellipses and {R₁, R₂, R₃.....} of rectangles as follows :

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

R₁ : rectangle of largest area, with sides parallel to the axes, inscribed in E₁;

En : ellipse $${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$$ of the 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?

In a non-right-angled triangle $$\Delta $$PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = $${\sqrt 3 }$$, q = 1, and the radius of the circumcircle of the $$\Delta $$PQR equals 1, then which of the following options is/are correct?

Let $$\alpha $$ and $$\beta $$ be the roots of$${x^2} - x - 1 = 0$$, with $$\alpha $$ > $$\beta $$. For all positive integers n, define

$${a_n} = {{{\alpha ^n} - {\beta ^n}} \over {\alpha - \beta }},\,n \ge 1$$

$${b_1} = 1\,and\,{b_n} = {a_{n - 1}} + {a_{n + 1}},\,n \ge 2$$

Then which of the following options is/are correct?

Let L₁ and L₂ denote the lines

$$r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)$$, $$\lambda $$$$ \in $$ R

and $$r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R$$

respectively. If L₃ is a line which is perpendicular to both L₁ and L₂ and cuts both of them, then which of the following options describe(s) L₃?