Let the circles

C₁ : x² + y² = 9 and C₂ : (x $$-$$ 3)² + (y $$-$$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $$-$$ h)² + (y $$-$$ k)² = r² satisfies the following conditions :

(i) Centre of C₃ is collinear with the centres of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃ and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$$\alpha $$y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only INCORRECT combination?

Let the circle C₁ : x² + y² = 9 and C₂ : (x $$-$$ 3)² + (y $$-$$ 4)² = 16, intersect at the points X and Y. Suppose that another circle C₃ : (x $$-$$ h)² + (y $$-$$ k)² = r² satisfies the following conditions :

(i) centre of C₃ is collinear with the centers of C₁ and C₂.

(ii) C₁ and C₂ both lie inside C₃, and

(iii) C₃ touches C₁ at M and C₂ at N.

Let the line through X and Y intersect C₃ at Z and W, and let a common tangent of C₁ and C₃ be a tangent to the parabola x² = 8$$\alpha $$y.

There are some expression given in the List-I whose values are given in List-II below.



Which of the following is the only CORRECT combination?

Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60$$^\circ $$ at one of its vertices N. Let the area of the $$\Delta $$LMN be $$4\sqrt 3 $$.

	List - I		List - II
P.	The length of the conjugate axis of H is	1.	8
Q.	The eccentricity of H is	2.	$${4 \over {\sqrt 3 }}$$
R.	The distance between the foci of H is	3.	$${2 \over {\sqrt 3 }}$$
S.	The length of the latus rectum of H is	4.	4

Let S be the circle in the XY-plane defined the equation x² + y² = 4.

Let E₁E₂ and F₁F₂ be the chords of S passing through the point P₀ (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G₁G₂ be the chord of S passing through P₀ and having slope$$-$$1. Let the tangents to S at E₁ and E₂ meet at E₃, then tangents to S at F₁ and F₂ meet at F₃, and the tangents to S at G₁ and G₂ meet at G₃. Then, the points E₃, F₃ and G₃ lie on the curve