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An angle of intersection of the curves, \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) and \( x^{2} + y^{2} = ab \), \( a > b \), is:

If the point on the curve \( y^{2}=6x \) nearest to the point \( \left(3,\dfrac{3}{2}\right) \) is \( (\alpha,\beta) \), then \( 2(\alpha+\beta) \) is equal to _____.

On the ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \) let \( P \) be a point in the second quadrant such that the tangent at \( P \) to the ellipse is perpendicular to the line \( x + 2y = 0 \). Let \( S \) and \( S' \) be the foci of the ellipse and \( e \) be its eccentricity. If \( A \) is the area of the triangle \( SPS' \) then, the value of \( (5 - e^2)A \) is:

Let \( T \) be the tangent to the ellipse \( E:\; x^{2}+4y^{2}=5 \) at the point \( P(1,1) \). If the area of the region bounded by the tangent \( T \), ellipse \( E \), lines \( x=1 \) and \( x=\sqrt{5} \) is \( \alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\!\left(\dfrac{1}{\sqrt{5}}\right) \), then \( \left|\alpha+\beta+\gamma\right| \) is equal to _______.

If \( x^{2} + 9y^{2} - 4x + 3 = 0, \, x, y \in \mathbb{R} \), then \( x \) and \( y \) respectively lie in the intervals:

A tangent line \( L \) is drawn at the point \( (2,-4) \) on the parabola \( y^{2}=8x \). If the line \( L \) is also tangent to the circle \( x^{2}+y^{2}=a \), then ‘\( a \)’ is equal to ______.

Let a line \( L: 2x + y = k,\ k > 0 \) be a tangent to the hyperbola \( x^2 - y^2 = 3 \). If \( L \) is also a tangent to the parabola \( y^2 = \alpha x \), then \( \alpha \) is equal to:

Let \( E \) be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at \( (3,-4) \), one focus at \( (4,-4) \) and one vertex at \( (5,-4) \). If \( mx-y=4, \ m>0 \) is a tangent to the ellipse \( E \), then the value of \( 5m^{2} \) is equal to ____________.

Consider the parabola with vertex \( \left( \frac{1}{2}, \frac{3}{4} \right) \) and the directrix \( y = \frac{1}{2} \). Let \( P \) be the point where the parabola meets the line \( x = -\frac{1}{2} \). If the normal to the parabola at \( P \) intersects the parabola again in the point \( Q \), then \( (PQ)^2 \) is equal to:

If two tangents drawn from a point \( P \) to the parabola \( y^{2}=16(x-3) \) are at right angles, then the locus of point \( P \) is:

Let \( A (\sec \theta, 2 \tan \theta) \) and \( B (\sec \phi, 2 \tan \phi) \), where \( \theta + \phi = \frac{\pi}{2} \), be two points on the hyperbola \( 2x^{2} - y^{2} = 2 \). If \( (\alpha, \beta) \) is the point of the intersection of the normals to the hyperbola at \( A \) and \( B \), then \( (2\beta)^{2} \) is equal to ________.

The length of the latus rectum of a parabola, whose vertex and focus are on the positive \( x \)-axis at a distance \( R \) and \( S \) (\( > R \)) respectively from the origin, is:

The line \( 12x \cos \theta + 5y \sin \theta = 60 \) is tangent to which of the following curves?

If the points of intersection of the ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 \) and the circle \( x^2 + y^2 = 4b,\, b>4 \) lie on the curve \( y^2 = 3x^2 \), then \( b \) is equal to:

If \(P\) is a point on the parabola \(y=x^{2}+4\) which is closest to the straight line \(y=4x-1\), then the co-ordinates of \(P\) are:

Let \( E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, \ a > b. \) Let \( E_2 \) be another ellipse such that it touches the end points of major axis of \( E_1 \) and the foci of \( E_2 \) are the end points of minor axis of \( E_1 \). If \( E_1 \) and \( E_2 \) have same eccentricities, then its value is:

The locus of the centroid of the triangle formed by any point \( P \) on the hyperbola \( 16x^{2}-9y^{2}+32x+36y-164=0 \) and its foci is:

Let \( L \) be a tangent to the parabola \( y^{2} = 4x - 20 \) at \( (6, 2) \). If \( L \) is also a tangent to the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{b} = 1 \), then the value of \( b \) is equal to:

Consider a hyperbola \( H : {x^{2}} - {2y^{2}} = 4 \). Let the tangent at a point \( P(4,\sqrt{6}) \) meet the x-axis at \( Q \) and latus rectum at \( R(x_{1},y_{1}), x_{1} > 0 \). If \( F \) is a focus of \( H \) which is nearer to the point \( P \), then the area of \( \triangle QFR \) is equal to:

Let an ellipse \( E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a^{2}>b^{2} \), passes through \( \left(\dfrac{\sqrt{3}}{2},1\right) \) and has eccentricity \( \dfrac{1}{\sqrt{3}} \). If a circle, centered at focus \( F(\alpha,0),\ \alpha>0 \) of \( E \) and radius \( \dfrac{2}{\sqrt{3}} \) intersects \( E \) at two points \( P \) and \( Q \), then \( PQ^{2} \) is equal to:

For which of the following curves, the line \(x+\sqrt{3}\,y=2\sqrt{3}\) is the tangent at the point \(\left(\dfrac{3\sqrt{3}}{2},\dfrac{1}{2}\right)\)?

A square \( ABCD \) has all its vertices on the curve \( x^2 y^2 = 1 \). The midpoints of its sides also lie on the same curve. Then, the square of area of \( ABCD \) is ______________ .

Let a parabola \( P \) be such that its vertex and focus lie on the positive \( x \)-axis at a distance \( 2 \) and \( 4 \) units from the origin, respectively. If tangents are drawn from \( O(0,0) \) to the parabola \( P \) which meet \( P \) at \( S \) and \( R \), then the area (in sq. units) of \( \triangle S O R \) is equal to:

Let \( y=mx+c,\; m>0 \) be the focal chord of \( y^{2}=-64x \), which is tangent to \( (x+10)^{2}+y^{2}=4 \). Then, the value of \( 4\sqrt{2}\,(m+c) \) is equal to _______.

If the minimum area of the triangle formed by a tangent to the ellipse \( \dfrac{x^2}{b^2} + \dfrac{y^2}{4a^2} = 1 \) and the co-ordinate axis is \( kab \), then \( k \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

Let a tangent be drawn to the ellipse \( \dfrac{x^{2}}{27} + y^{2} = 1 \) at \( \left(3\sqrt{3}\cos\theta,\ \sin\theta\right) \) where \( \theta \in \left(0,\ \dfrac{\pi}{2}\right) \). Then the value of \( \theta \) such that the sum of intercepts on axes made by this tangent is minimum is equal to:

A tangent and a normal are drawn at the point \( P(2,\,-4) \) on the parabola \( y^2 = 8x \), which meet the directrix of the parabola at the points \( A \) and \( B \) respectively. If \( Q(a,\, b) \) is a point such that \( AQBP \) is square, then \( 2a + b \) is equal to:

If the area of the triangle formed by the positive \( x \)-axis, the normal and the tangent to then circle \( (x-2)^2 + (y-3)^2 = 25 \) at the point \( (5,7) \) is \( A \), then \( 24A \) is equal to __________.

The locus of mid-points of the line segments joining \( (-3,-5) \) and the points on the ellipse \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \) is:

Let \( C \) be the locus of the mirror image of a point on the parabola \( y^{2}=4x \) with respect to the line \( y=x \). Then the equation of tangent to \( C \) at \( P(2,1) \) is:

Let \( \theta \) be the acute angle between the tangents to the ellipse \( \frac{x^2}{9} + \frac{y^2}{1} = 1 \) and the circle \( x^2 + y^2 = 3 \) at their point of intersection in the first quadrant. Then \( \tan \theta \) is equal to:

Let the tangent to the parabola \( S: y^{2}=2x \) at the point \( P(2,2) \) meet the \( x \)-axis at \( Q \) and normal at it meet the parabola \( S \) at the point \( R \). Then the area (in sq. units) of the triangle \( PQR \) is equal to:

If a tangent to the ellipse \( {x^2} + 4y^2 = 4 \) meets the tangents at the extremities of its major axis at \( B \) and \( C \), then the circle with \( BC \) as diameter passes through the point:

The point \( P(-2\sqrt{6}, \sqrt{3}) \) lies on the hyperbola \( \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) having eccentricity \( \dfrac{\sqrt{5}}{2} \). If the tangent and normal at \( P \) to the hyperbola intersect its conjugate axis at the points \( Q \) and \( R \) respectively, then \( QR \) is equal to:

The locus of the mid points of the chords of the hyperbola \( x^2 - y^2 = 4 \), which touch the parabola \( y^2 = 8x \), is :

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JEE Main 2021 27th August Morning Shift

If \( x^{2} + 9y^{2} - 4x + 3 = 0, \, x, y \in \mathbb{R} \), then \( x \) and \( y \) respectively lie in the intervals: