Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is

Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the
parabola y² = 4$$\lambda $$x, and suppose the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

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?