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?

Consider two straight lines, each of which is tangent to both the circle x² + y² = (1/2) and the parabola y² = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is $$\sqrt 2 $$, then which of the following statement(s) is (are) TRUE?

If $$2x - y + 1 = 0$$ is a tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {16}} = 1$$ then which of the following CANNOT be sides of a right angled triangle?

If a chord, which is not a tangent, of the parabola y² = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) possible value(s) of p, h and k?