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If the length of the latus rectum of the ellipse \( x^{2}+4y^{2}+2x+8y-\lambda=0 \) is \( 4 \), and \( l \) is the length of its major axis, then \( \lambda + l \) is equal to _______.

If the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \) meets the line \( \dfrac{x}{7}+\dfrac{y}{2\sqrt{6}}=1 \) on the x-axis and the line \( \dfrac{x}{7}-\dfrac{y}{2\sqrt{6}}=1 \) on the y-axis, then the eccentricity of the ellipse is:

Let the focal chord of the parabola \(P: y^{2}=4x\) along the line \(L: y=mx+c,\, m>0\) meet the parabola at the points \(M\) and \(N\). Let the line \(L\) be a tangent to the hyperbola \(H: x^{2}-y^{2}=4\). If \(O\) is the vertex of \(P\) and \(F\) is the focus of \(H\) on the positive \(x\)-axis, then the area of the quadrilateral \(OMFN\) is:

Let \( PQ \) be a focal chord of the parabola \( y^2 = 4x \) such that it subtends an angle of \( \pi/2 \) at the point \( (3, 0) \). Let the line segment \( PQ \) be also a focal chord of the ellipse \( E : \, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \, a^2 > b^2 \). If \( e \) is the eccentricity of the ellipse \( E \), then the value of \( \frac{1}{e^2} \) is equal to:

Let a line \( L \) pass through the point of intersection of the lines \( bx+10y-8=0 \) and \( 2x-3y=0, \, b\in \mathbb{R}-\left\{ \dfrac{4}{3} \right\} \). If the line \( L \) also passes through the point \( (1,1) \) and touches the circle \( 17(x^{2}+y^{2})=16 \), then the eccentricity of the ellipse \( \dfrac{x^{2}}{5}+\dfrac{y^{2}}{b^{2}}=1 \) is:

Let \( x = 2t,\, y = \frac{t^2}{3} \) be a conic. Let \( S \) be the focus and \( B \) be the point on the axis of the conic such that \( SA \perp BA \), where \( A \) is any point on the conic. If \( k \) is the ordinate of the centroid of \( \triangle SAB \), then \( \lim\limits_{t \to 1} k \) is equal to:

The normal to the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{9}=1 \) at the point \( (8,\,3\sqrt{3}) \) on it passes through the point:

Let \( P_{1} \) be parabola with vertex \( (3, 2) \) and focus \( (4, 4) \) and \( P_{2} \) be its mirror image with respect to the line \( x + 2y = 6 \). Then the directrix of \( P_{2} \) is \( x + 2y = \) ________ .

Let a line \( L_1 \) be tangent to the hyperbola \( \dfrac{x^2}{16} - \dfrac{y^2}{4} = 1 \) and let \( L_2 \) be the line passing through the origin and perpendicular to \( L_1 \). If the locus of the point of intersection of \( L_1 \) and \( L_2 \) is \( (x^2 + y^2)^2 = \alpha x^2 + \beta y^2 \), then \( \alpha + \beta \) is equal to _______.

Let the normal at the point \( P \) on the parabola \( y^2 = 6x \) pass through the point \( (5, -8) \). If the tangent at \( P \) to the parabola intersects its directrix at the point \( Q \), then the ordinate of the point \( Q \) is:

Let the tangent drawn to the parabola \( y^2 = 24x \) at the point \( (\alpha, \beta) \) is perpendicular to the line \( 2x + 2y = 5 \). Then the normal to the hyperbola \( \dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1 \) at the point \( (\alpha + 4, \beta + 4) \) does NOT pass through the point:

For real numbers \( a, b \) (\( a > b > 0 \)), let
\( \text{Area}\left\{(x,y): x^2 + y^2 \le a^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \ge 1 \right\} = 30\pi \) and \( \text{Area}\left\{(x,y): x^2 + y^2 \ge b^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \right\} = 18\pi \).
Then the value of \( (a-b)^2 \) is equal to ______.

A common tangent \( T \) to the curves \( C_{1}:\ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1 \) and \( C_{2}:\ \dfrac{x^{2}}{42}-\dfrac{y^{2}}{143}=1 \) does not pass through the fourth quadrant. If \( T \) touches \( C_{1} \) at \( (x_{1},y_{1}) \) and \( C_{2} \) at \( (x_{2},y_{2}) \), then \( |2x_{1}+x_{2}| \) is equal to ______.

If the length of the latus rectum of a parabola, whose focus is \( (a,a) \) and the tangent at its vertex is \( x+y=a \), is \( 16 \), then \( |a| \) is equal to:

Let the eccentricity of the hyperbola \( H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be \( \sqrt{\frac{5}{2}} \) and length of its latus rectum be \( 6\sqrt{2} \). If \( y = 2x + c \) is a tangent to the hyperbola \( H \), then the value of \( c^2 \) is equal to:

Let the common tangents to the curves \( 4(x^{2}+y^{2}) = 9 \) and \( y^{2} = 4x \) intersect at the point \( Q \). Let an ellipse, centered at the origin \( O \), has lengths of semi-minor and semi-major axes equal to \( OQ \) and \( 6 \), respectively. If \( e \) and \( l \) respectively denote the eccentricity and the length of the latus rectum of this ellipse, then \( \frac{l}{e^{2}} \) is equal to______________.

If two tangents drawn from a point \( (\alpha, \beta) \) lying on the ellipse \( 25x^{2} + 4y^{2} = 1 \) to the parabola \( y^{2} = 4x \) are such that the slope of one tangent is four times the other, then the value of \( (10\alpha + 5)^{2} + (16\beta^{2} + 50)^{2} \) equals ________________.

Let the tangents at the points \( P \) and \( Q \) on the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{4} = 1 \) meet at the point \( R(\sqrt{2},\, 2\sqrt{2} - 2) \). If \( S \) is the focus of the ellipse on its negative major axis, then \( SP^{2} + SQ^{2} \) is equal to __________________.

An ellipse \( E:\; \displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) passes through the vertices of the hyperbola \( H:\; \displaystyle\frac{x^{2}}{49}-\frac{y^{2}}{64}=-1 \). Let the major and minor axes of the ellipse \( E \) coincide with the transverse and conjugate axes of the hyperbola \( H \), respectively. Let the product of the eccentricities of \( E \) and \( H \) be \( {1}/{2} \). If \( l \) is the length of the latus rectum of the ellipse \( E \), then the value of \( 113l \) is equal to ______.

Let \( P(a,b) \) be a point on the parabola \( y^{2}=8x \) such that the tangent at \( P \) passes through the centre of the circle \( x^{2}+y^{2}-10x-14y+65=0 \). Let \( A \) be the product of all possible values of \( a \) and \( B \) be the product of all possible values of \( b \). Then the value of \( A+B \) is equal to:

The tangents at the points \( A(1, 3) \) and \( B(1, -1) \) on the parabola \( y^2 - 2x - 2y = 1 \) meet at the point \( P \). Then the area (in \( \text{unit}^2 \)) of the triangle \( PAB \) is:

Two tangent lines \(l_1\) and \(l_2\) are drawn from the point \( (2, 0) \) to the parabola \( 2y^2 = -x \). If the lines \(l_1\) and \(l_2\) are also tangent to the circle \( (x-5)^2 + y^2 = r \), then \( 17r \) is equal to _______________.

If \( m \) is the slope of a common tangent to the curves \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1 \) and \( x^{2} + y^{2} = 12 \), then \( 12 m^{2} \) is equal to:

Let the hyperbola \( H : \frac{x^{2}}{a^{2}} - y^{2} = 1 \) and the ellipse \( E : 3x^{2} + 4y^{2} = 12 \) be such that the length of latus rectum of \( H \) is equal to the length of latus rectum of \( E \). If \( e_{H} \) and \( e_{E} \) are the eccentricities of \( H \) and \( E \) respectively, then the value of \( 12\left(e_{H}^{2} + e_{E}^{2}\right) \) is equal to ________ .

If the equation of the parabola, whose vertex is at \( (5,4) \) and the directrix is \( 3x + y - 29 = 0 \), is \( x^{2} + a y^{2} + bxy + cx + dy + k = 0 \), then \( a + b + c + d + k \) is equal to

The area (in sq. units) of the region enclosed between the parabola \( y^{2} = 2x \) and the line \( x + y = 4 \) is ________ .

Let \( l \) be a line which is normal to the curve \( y = 2x^2 + x + 2 \) at a point \( P \) on the curve. If the point \( Q(6, 4) \) lies on the line \( l \) and \( O \) is origin, then the area of the triangle \( OPQ \) is equal to ________.

Let \( \lambda x - 2y = \mu \) be a tangent to the hyperbola \( a^{2}x^{2} - y^{2} = b^{2} \). Then \( \left( \frac{\lambda}{a} \right)^{2} - \left( \frac{\mu}{b} \right)^{2} \) is equal to:

Let the foci of the ellipse \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{7} = 1 \) and the hyperbola \( \dfrac{x^{2}}{144} - \dfrac{y^{2}}{\alpha} = \dfrac{1}{25} \) coincide. Then the length of the latus rectum of the hyperbola is:

Let \( H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 , \ a>0 , \ b>0 \) be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is \( 4\left(2\sqrt{2}+\sqrt{14}\right) \). If the eccentricity \( H \) is \( \dfrac{\sqrt{11}}{2} \), then the value of \( a^2+b^2 \) is equal to __________.

Let the hyperbola \( H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) pass through the point \( (2\sqrt{2}, -2\sqrt{2}) \). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is \( e \) times the length of the latus rectum of H, where \( e \) is the eccentricity of H, then which of the following points lies on the parabola

Let \( P: y^{2}=4ax,\; a>0 \) be a parabola with focus \( S \). Let the tangents to the parabola \( P \) make an angle of \( \frac{\pi}{4} \) with the line \( y=3x+5 \) touch the parabola \( P \) at \( A \) and \( B \). Then the value of \( a \) for which \( A \), \( B \) and \( S \) are collinear is:

If \( y=m_{1}x+c_{1} \) and \( y=m_{2}x+c_{2},\ m_{1}\neq m_{2} \) are two common tangents of circle \( x^{2}+y^{2}=2 \) and parabola \( y^{2}=x \), then the value of \( 8|\,m_{1}m_{2}\,| \) is equal to:

Let the equation of two diameters of a circle \( x^2 + y^2 - 2x + 2fy + 1 = 0 \) be \( 2px - y = 1 \) and \( 2x + py = 4p \). Then the slope \( m \in (0,\infty) \) of the tangent to the hyperbola \( 3x^2 - y^2 = 3 \) passing through the centre of the circle is equal to ________.

Let the eccentricity of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), be \( \frac{1}{4} \). If this ellipse passes through the point \( \left(-4\sqrt{\frac{2}{5}},\, 3\right) \), then \( a^2 + b^2 \) is equal to:

Let \( a>0, b>0 \). Let \( e \) and \( l \) respectively be the eccentricity and length of the latus rectum of the hyperbola \( \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \). Let \( e' \) and \( l' \) respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If \( e^2 = \dfrac{11}{14}\,l \) and \( (e')^2 = \dfrac{11}{8}\,l' \), then the value of \( 77a + 44b \) is equal to:

The locus of the mid point of the line segment joining the point \( (4, 3) \) and the points on the ellipse \( x^2 + 2y^2 = 4 \) is an ellipse with eccentricity:

Let the mirror image of a circle \( c_1:x^2+y^2-2x-6y+\alpha=0 \) in line \( y=x+1 \) be \( c_2:5x^2+5y^2+10gx+10fy+38=0 \). If \( r \) is the radius of circle \( c_2 \), then \( \alpha+6r^2 \) is equal to__________.

Let the maximum area of the triangle that can be inscribed in the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1 \), \( a > 2 \), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the \( y \)-axis, be \( 6\sqrt{3} \). Then the eccentricity of the ellipse is:

The line \( y = x + 1 \) meets the ellipse \( \frac{x^2}{4} + \frac{y^2}{2} = 1 \) at two points \( P \) and \( Q \). If \( r \) is the radius of the circle with \( PQ \) as diameter then \( (3r)^2 \) is equal to:

A circle of radius \( 2 \) unit passes through the vertex and the focus of the parabola \( y^{2} = 2x \) and touches the parabola \( y = \left( x - \dfrac{1}{4} \right)^{2} + \alpha \), where \( \alpha > 0 \). Then \( (4\alpha - 8)^{2} \) is equal to _______.

The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x-2)^2 \) is:

Let \( S=\{(x,y)\in \mathbb{N}\times \mathbb{N}:9(x-3)^2+16(y-4)^2\leq 144\} \) and \(\displaystyle T = \{(x, y) \in \mathbb{R} \times \mathbb{R} : (x - 7)^2 + (y - 4)^2 \le 36\}\). Then \(n(S\cap T)\) is equal to __________.

If vertex of a parabola is \( (2,-1) \) and the equation of its directrix is \( 4x - 3y = 21 \), then the length of its latus rectum is:

Let the eccentricity of the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be \( \frac{5}{4} \). If the equation of the normal at the point \( \left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right) \) on the hyperbola is \( 8\sqrt{5}\,x+\beta y=\lambda \), then \( \lambda-\beta \) is equal to __________.

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JEE Main 2022 29th July Evening shift

Let \( S=\{(x,y)\in \mathbb{N}\times \mathbb{N}:9(x-3)^2+16(y-4)^2\leq 144\} \) and \(\displaystyle T = \{(x, y) \in \mathbb{R} \times \mathbb{R} : (x - 7)^2 + (y - 4)^2 \le 36\}\). Then \(n(S\cap T)\) is equal to __________.