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Let \( T_1 \) and \( T_2 \) be two distinct common tangents to the ellipse \( E: \, \frac{x^2}{6} + \frac{y^2}{3} = 1 \) and the parabola \( P: \, y^2 = 12x \). Suppose that the tangent \( T_1 \) touches \( P \) and \( E \) at the points \( A_1 \) and \( A_2 \), respectively and the tangent \( T_2 \) touches \( P \) and \( E \) at the points \( A_4 \) and \( A_3 \), respectively. Then which of the following statements is(are) true?

Let \( T \) be the tangent to the parabola \( y^{2}=16x \) at the point \( (64,\,32) \). Let \( L \) be the tangent to the same parabola at another point \( (x_{1},y_{1}) \) on the parabola. If \( L \) and \( T \) are perpendicular to each other, then the distance between the point \( (x_{1},y_{1}) \) and the focus of the parabola, is:

Consider the ellipse \( E \) given by \( \frac{x^2}{18} + \frac{y^2}{12} = 1 \). Let \( H \) be the hyperbola whose eccentricity is the reciprocal of the eccentricity of \( E \) and whose foci are the same as that of \( E \). Let \( P \) and \( Q \) be the points of intersection of \( H \) and the parabola \( \sqrt{5}\,y = x^2 \) in the first quadrant. Let \( d \) be the distance between \( P \) and \( Q \).

Let \( S \) denote the locus of the point of intersection of the pair of lines
\( 4x - 3y = 12\alpha, \)
\( 4\alpha x + 3\alpha y = 12 \),
where \( \alpha \) varies over the set of non-zero real numbers. Let \( T \) be the tangent to \( S \) passing through the points \( (p,0) \) and \( (0,q) \), \( q > 0 \), and parallel to the line \( 4x - \dfrac{3}{\sqrt{2}} y = 0 \).
Then the value of \( pq \) is

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Consider the ellipses given by \( x^2 + 4y^2 = 1 \) and \( 4x^2 + y^2 = 1 \).
If \( \alpha \) is the area of the common region that lies inside both the given ellipses, then the value of \( \cot \alpha \) is ____________.

A normal with slope \( \frac{1}{\sqrt{6}} \) is drawn from the point \( (0, -\alpha) \) to be parabola \( x^{2} = -4ay \), where \( a > 0 \). Let \( L \) be the line passing through \( (0, -\alpha) \) and parallel to the directrix of the parabola. Suppose that \( L \) intersects the parabola at two points \( A \) and \( B \). Let \( r \) denote the length of the latus rectum and \( s \) denote the square of the length of the line segment \( AB \). If \( r : s = 1 : 16 \) then the value of \( 24a \) is __________.

Let \( P(x_1,y_1) \) and \( Q(x_2,y_2) \) be two distinct points on the ellipse \( \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1 \) such that \( y_1 > 0 \) and \( y_2 > 0 \). Let \( C \) denote the circle \( x^2 + y^2 = 9 \), and \( M \) be the point \( (3, 0) \). Suppose the line \( x = x_1 \) intersects \( C \) at \( R \), and the line \( x = x_2 \) intersects \( C \) at \( S \), such that the \( y \)-coordinates of \( R \) and \( S \) are positive.
Let \( \angle ROM = \dfrac{\pi}{6} \) and \( \angle SOM = \dfrac{\pi}{3} \), where \( O \) denotes the origin \( (0, 0) \). Let \( |XY| \) denote the length of the line segment \( XY \).
Then which of the following statements is (are) TRUE? 

Let \( A_1,B_1,C_1 \) be three points in the \( xy \)-plane. Suppose that the lines \( A_1C_1 \) and \( B_1C_1 \) are tangents to the curve \( y^2=8x \) at \( A_1 \) and \( B_1 \), respectively. If \( O=(0,0) \) and \( C_1=(-4,0) \), then which of the following statements is (are) TRUE?

Let \( S \) denote the locus of the mid-points of those chords of the parabola \( y^{2} = x \), such that the area of the region enclosed between the parabola and the chord is \( \dfrac{4}{3} \). Let \( R \) denote the region lying in the first quadrant, enclosed by the parabola \( y^{2} = x \), the curve \( S \), and the lines \( x = 1 \) and \( x = 4 \).
Then which of the following statements is (are) TRUE?

Consider the region \( R = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x \geq 0 \text{ and } y^2 \leq 4 - x\} \). Let \( F \) be the family of all circles that are contained in \( R \) and have centers on the \( x \)-axis. Let \( C \) be the circle that has largest radius among the circles in \( F \). Let \( (\alpha, \beta) \) be a point where the circle \( C \) meets the curve \( y^2 = 4 - x \).
The radius of the circle \( C \) is ______.

Let \( E \) denote the parabola \( y^{2}=8x \). 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?

Consider the parabola \( y^2=4x \). Let \( S \) be the focus of the parabola. A pair of tangents drawn to the parabola from the point \( P=(-2,1) \) meet the parabola at \( P_1 \) and \( P_2 \). Let \( Q_1 \) and \( Q_2 \) be points on the lines \( SP_1 \) and \( SP_2 \) respectively such that \( PQ_1 \) is perpendicular to \( SP_1 \) and \( PQ_2 \) is perpendicular to \( SP_2 \). Then, which of the following is/are TRUE?

Let \( E \) be the ellipse \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \). For any three distinct points \( P, Q \) and \( Q' \) on \( E \), let \( M(P, Q) \) be the mid-point of the line segment joining \( P \) and \( Q \), and \( M(P, Q') \) be the mid-point of the line segment joining \( P \) and \( Q' \). Then the maximum possible value of the distance between \( M(P, Q) \) and \( M(P, Q') \), as \( P, Q \) and \( Q' \) vary on \( E \), is \( \_\_\_\_\_\_\_ \).

Consider the ellipse \( \dfrac{x^{2}}{9} + \dfrac{y^{2}}{4} = 1 \). \( S(p, q) \) be a point in the first quadrant such that \( \dfrac{p^{2}}{9} + \dfrac{q^{2}}{4} > 1 \). Two tangents are drawn from \( S \) to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point \( T \) in the fourth quadrant. Let \( R \) be the vertex of the ellipse with positive \( x \)-coordinate and \( O \) be the center of the ellipse. If the area of the triangle \( \Delta ORT \) is \( \dfrac{3}{2} \), then which of the following options is correct?

Let \( P \) be a point on the parabola \( y^2=4ax \), where \( a>0 \). The normal to the parabola at \( P \) meets the \( x \)-axis at a point \( Q \). The area of the triangle \( PFQ \), 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: