JEE Mains JEE Advance BITSAT
JEE Mains | PYQ

Year

All years 2026 2025 2024 2023 2022 2021

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 __________.

← PreviousQuestion 37 of 45Next →

JEE Main 2022 26th June Evening Shift

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: