Let E denote the parabola y² = 8x. Let P = ($$-$$2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let $$\Delta $$ denote the area of the triangle formed by the tangent at P, the normal at P and the X-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

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?

Let T be the line passing through the points P($$-$$2, 7) and Q(2, $$-$$5). Let F₁ be the set of al pairs of circles (S₁, S₂) such that T is tangent to S₁ at P and tangent to S₂ at Q, and also such that S₁ and S₂ touch each other at a point, say M. Let E₁ be the set representing the locus of M as the pair (S₁, S₂) varies in F₁. Let the set of all straight line segments joining a pair of distinct points of E₁ and passing through the point R(1, 1) be F₂. Let E₂ be the set of the mid-points of the line segments in the set F₂. Then, which of the following statement(s) is (are) TRUE?