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Consider ellipses \( E_k : kx^2 + k^2y^2 = 1, \, k = 1, 2, \ldots, 20 \). Let \( C_k \) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \( E_k \). If \( r_k \) is the radius of the circle \( C_k \), then the value of \( \displaystyle \sum_{k=1}^{20} \frac{1}{r_k^2} \) is:

Let \( y=p(x) \) be the parabola passing through the points \( (-1,0), (0,1) \) and \( (1,0) \). If the area of the region \( \{(x,y):(x+1)^2+(y-1)^2\le 1,\; y\le p(x)\} \) is \( A \), then \( 12(\pi-4A) \) is equal to ________.

Let the eccentricity of an ellipse \( \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \) is reciprocal to that of the hyperbola \( 2x^{2}-2y^{2}=1 \). If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse ________ .

Let \( H_{n}:\ \dfrac{x^{2}}{1+n}-\dfrac{y^{2}}{3+n}=1,\ n\in\mathbb{N} \). Let \( k \) be the smallest even value of \( n \) such that the eccentricity of \( H_{k} \) is a rational number. If \( l \) is the length of the latus rectum of \( H_{k} \), then \( 21l \) is equal to __________ .

Let \( H \) be the hyperbola, whose foci are \( (1 \pm \sqrt{2}, 0) \) and eccentricity is \( \sqrt{2} \). Then the length of its latus rectum is:

The distance of the point \( (6,-2\sqrt{2}) \) from the common tangent \( y=mx+c,\, m>0 \), of the curves \( x=2y^{2} \) and \( x=1+y^{2} \) is:

If the radius of the largest circle with centre \( (2,0) \) inscribed in the ellipse \( x^2 + 4y^2 = 36 \) is \( r \), then \( 12r^2 \) is equal to:

If the \( x \)-intercept of a focal chord of the parabola \( y^2 = 8x + 4y + 4 \) is \( 3 \), then the length of this chord is equal to \( \_\_\_\_ \).

The equations of two sides of a variable triangle are \( x=0 \) and \( y=3 \), and its third side is a tangent to the parabola \( y^{2}=6x \). The locus of its circumcentre is:

Let \( A(0,1), B(1,1) \) and \( C(1,0) \) be the mid-points of the sides of a triangle with incentre at the point \( D \). If the focus of the parabola \( y^{2}=4ax \) passing through \( D \) is \( (\alpha+\beta\sqrt{2},0) \), where \( \alpha \) and \( \beta \) are rational numbers, then \( \dfrac{\alpha}{\beta^{2}} \) is equal to:

Let \( P\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), Q, R \) and \( S \) be four points on the ellipse \( 9x^2 + 4y^2 = 36 \). Let \( PQ \) and \( RS \) be mutually perpendicular and pass through the origin. If \( \dfrac{1}{(PQ)^2} + \dfrac{1}{(RS)^2} = \dfrac{p}{q} \), where \( p \) and \( q \) are coprime, then \( p+q \) is equal to:

Let the tangent to the parabola \( y^{2}=12x \) at the point \( (3,\alpha) \) be perpendicular to the line \( 2x+2y=3 \). Then the square of distance of the point \( (6,-4) \) from the normal to the hyperbola \( \alpha^{2}x^{2}-9y^{2}=9\alpha^{2} \) at its point \( (\alpha-1,\alpha+2) \) is equal to ______.

The line \( x = 8 \) is the directrix of the ellipse \( E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with the corresponding focus \( (2, 0) \). If the tangent to \( E \) at the point \( P \) in the first quadrant passes through the point \( (0, 4\sqrt{3}) \) and intersects the \( x \)-axis at \( Q \), then \( (3PQ)^{2} \) is equal to ______.

Let the tangent and normal at the point \( (3\sqrt{3}, 1) \) on the ellipse \( \dfrac{x^2}{36} + \dfrac{y^2}{4} = 1 \) meet the y-axis at the points \( A \) and \( B \) respectively. Let the circle \( C \) be drawn taking \( AB \) as a diameter and the line \( x = 2\sqrt{5} \) intersect \( C \) at the points \( P \) and \( Q \). If the tangents at the points \( P \) and \( Q \) on the circle intersect at the point \( (\alpha, \beta) \), then \( \alpha^2 - \beta^2 \) is equal to:

The vertices of a hyperbola \( H \) are \( (\pm 6, 0) \) and its eccentricity is \( \frac{\sqrt{5}}{2} \). Let \( N \) be the normal to \( H \) at a point in the first quadrant and parallel to the line \( \sqrt{2}x + y = 2\sqrt{2} \). If \( d \) is the length of the line segment of \( N \) between \( H \) and the \( y \)-axis then \( d^2 \) is equal to ______ .

In a group of \( 100 \) persons \( 75 \) speak English and \( 40 \) speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is \( \alpha \) and the number of persons who speak only Hindi is \( \beta \), then the eccentricity of the ellipse \( 25(\beta^{2}x^{2}+a^{2}y^{2})=\alpha^{2}\beta^{2} \) is:

If the tangent at a point \( P \) on the parabola \( y^{2}=3x \) is parallel to the line \( x+2y=1 \) and the tangents at the points \( Q \) and \( R \) on the ellipse \( \dfrac{x^{2}}{4}+\dfrac{y^{2}}{1}=1 \) are perpendicular to the line \( x-y=2 \), then the area of the triangle \( PQR \) is:

Let \( T \) and \( C \) respectively be the transverse and conjugate axes of the hyperbola \( 16x^{2}-y^{2}+64x+4y+44=0 \). Then the area of the region above the parabola \( x^{2}=y+4 \), below the transverse axis \( T \) and on the right of the conjugate axis \( C \) is:

Let \( R \) be the focus of the parabola \( y^2 = 20x \) and the line \( y = mx + c \) intersect the parabola at two points \( P \) and \( Q \). Let the point \( G(10, 10) \) be the centroid of the triangle \( PQR \). If \( c - m = 6 \), then \( (PQ)^2 \) is:

The foci of a hyperbola are \( (\pm 2, 0) \) and its eccentricity is \( \dfrac{3}{2} \). A tangent, perpendicular to the line \( 2x + 3y = 6 \), is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x and y-axes are \( a \) and \( b \) respectively, then \( |6a| + |5b| \) is equal to _________.

Let a circle of radius \( 4 \) be concentric to the ellipse \( 15x^2+19y^2=285 \). Then the common tangents are inclined to the minor axis of the ellipse at the angle.

A circle with centre \( (2, 3) \) and radius \( 4 \) intersects the line \( x + y = 3 \) at the points \( P \) and \( Q \). If the tangents at \( P \) and \( Q \) intersect at the point \( S(\alpha, \beta) \), then \( 4\alpha - 7\beta \) is equal to _________.

Let a tangent to the curve \( y^{2}=24x \) meet the curve \( xy=2 \) at the points \( A \) and \( B \). Then the mid-points of such line segment \( AB \) lie on a parabola with the:

If the maximum distance of normal to the ellipse \( \frac{x^2}{4} + \frac{y^2}{b^2} = 1, \; b < 2 \), from the origin is \( 1 \), then the eccentricity of the ellipse is:

Let \( P(x_{0},y_{0}) \) be the point on the hyperbola \( 3x^{2}-4y^{2}=36 \), which is nearest to the line \( 3x+2y=1 \). Then \( \sqrt{2}\,(y_{0}-x_{0}) \) is equal to:

Let a tangent to the curve \( 9x^{2} + 16y^{2} = 144 \) intersect the coordinate axes at the points \( A \) and \( B \). Then, the minimum length of the line segment \( AB \) is _______.

Let \( R \) be a rectangle given by the lines \( x=0 \), \( x=2 \), \( y=0 \) and \( y=5 \). Let \( A(\alpha, 0) \) and \( B(0, \beta) \), \( \alpha \in [0, 2] \) and \( \beta \in [0, 5] \), be such that the line segment \( AB \) divides the area of the rectangle \( R \) in the ratio \( 4:1 \). Then, the mid-point of \( AB \) lies on a :

Let \( C \) be the largest circle centred at \( (2, 0) \) and inscribed in the ellipse \( \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 \). If \( (1, \alpha) \) lies on \( C \), then \( 10\alpha^{2} \) is equal to ______.

The ordinates of the points \( P \) and \( Q \) on the parabola with focus \( (3,0) \) and directrix \( x=-3 \) are in the ratio \( 3:1 \). If \( R(\alpha,\beta) \) is the point of intersection of the tangents to the parabola of \( P \) and \( Q \), then \( \dfrac{\beta^2}{\alpha} \) is equal to ________.

Let \( PQ \) be a focal chord of the parabola \( y^{2} = 36x \) of length \( 100 \), making an acute angle with the positive \( x \)-axis. Let the ordinate of \( P \) be positive and \( M \) be the point on the line segment \( PQ \) such that \( PM : MQ = 3 : 1 \). Then which of the following points does NOT lie on the line passing through \( M \) and perpendicular to the line \( PQ \)?

A triangle is formed by the tangents at the point \( (2, 2) \) on the curves \( y^2 = 2x \) and \( x^2 + y^2 = 4x \), and the line \( x + y + 2 = 0 \). If \( r \) is the radius of its circumcircle, then \( r^2 \) is equal to ________.

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JEE Main 2023 24th Jan Morning Shift

Let a tangent to the curve \( y^{2}=24x \) meet the curve \( xy=2 \) at the points \( A \) and \( B \). Then the mid-points of such line segment \( AB \) lie on a parabola with the: