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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Conic Section PYQ

JEE MAIN PYQ
The area of the smaller region enclosed by the curves $y^2 = 8x + 4$ and $x^2 + y^2 + 4\sqrt{3}x - 4 = 0$ is equal to





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

JEE MAIN PYQ
The equation of the circle passing through the foci of the ellipse $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{9}=1$, and having centre at $(0,3)$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\dfrac{x^2}{b^2}+\dfrac{y^2}{25}=1$ and the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{b^2}=1$, respectively. If $b<5$ and $e_1e_2=1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\sqrt {{3 \over 2}} ,1} \right)$ and has eccentricity ${1 \over {\sqrt 3 }}$. If a circle, centered at focus F($\alpha$$, 0), $\alpha$$ > 0, of E and radius ${2 \over {\sqrt 3 }}$, intersects E at two points P and Q, then PQ2 is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the line x $-$ y = 1 and the curve x2 = 2y is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

JEE MAIN PYQ
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?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let the ellipse $E:{x^2} + 9{y^2} = 9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is ${m \over n}$, where m and n are coprime, then $m - n$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

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





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Morning Shift) PYQ

Solution

JEE MAIN PYQ
If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
The area of the region bounded by y2 = 8x and y2 = 16(3 $-$ x) is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

JEE MAIN PYQ
If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the point $(\alpha, \dfrac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the lines $x\cos\theta + y\sin\theta = 7, \theta \in (0, \dfrac{\pi}{2})$ between the co-ordinates axes, then $\alpha$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $\mathrm{P}\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9x^{2}+4y^{2}=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ 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:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
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:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
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 $PQ$?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the foci of a hyperbola are the same as those of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{25}=1$ and the eccentricity of the hyperbola is $\dfrac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2},\ \dfrac{14}{3}\sqrt{\dfrac{2}{5}}\right)$ on the hyperbola is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $a>b$, of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of ${\pi \over 2}$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$. If e is the eccentricity of the ellipse E, then the value of ${1 \over {{e^2}}}$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2–1 below the x-axis, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

JEE MAIN PYQ
A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{25} = 1$ at $A$ and $B$ such that $(PA) \cdot (PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\ a > b$ be an ellipse, whose eccentricity is $\dfrac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the **square of the eccentricity** of $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
For $0 < \theta < \dfrac{\pi}{2}$, if the eccentricity of the hyperbola $x^2 - y^2 \csc^2\theta = 5$ is $\sqrt{7}$ times the eccentricity of the ellipse $x^2 \csc^2\theta + y^2 = 5,$ then the value of $\theta$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let the eccentricity of the ellipse ${x^2} + {a^2}{y^2} = 25{a^2}$ be b times the eccentricity of the hyperbola ${x^2} - {a^2}{y^2} = 5$, where a is the minimum distance between the curves y = ex and y = logex. Then ${a^2} + {1 \over {{b^2}}}$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (30 June Morning Shift) PYQ

Solution

JEE MAIN PYQ
The shortest distance between the curves $y^2=8x$ and $x^2+y^2+12y+35=0$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

JEE MAIN PYQ
If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Morning) PYQ

Solution

JEE MAIN PYQ
Let $P$ be a point on the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$. Let the line passing through $P$ and parallel to the $y$–axis meet the circle $x^{2}+y^{2}=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$–axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ (as $P$ moves on the ellipse) is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

JEE MAIN PYQ
A hyperbola whose transverse axis is along the major axis of the conic $\dfrac{x^2}{3} + \dfrac{y^2}{4} = 4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\dfrac{3}{2}$, then which of the following points does NOT lie on it?





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$ and $H:\ \dfrac{x^{2}}{A^{2}}-\dfrac{y^{2}}{B^{2}}=1$. Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a-A=2$, and the ratio of the eccentricities of $E$ and $H$ is $\dfrac{1}{3}$, then the sum of the lengths of their latus recta is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

JEE MAIN PYQ
A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is $14$. Let $f(x,y)=0$ be the locus of $P$, which intersects the $x$-axis at the points $A,B$ and the $y$-axis at the points $C,D$. Then the area of the quadrilateral $ACBD$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution


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