JEE MAIN Ellipse Previous Year Question Paper and Solution with PDF

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Previous Year Questions

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Previous Year Questions
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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Ellipse PYQ

Qus : 1 JEE MAIN PYQ

The length of the chord of the ellipse $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{2}=1$ whose midpoint is $\left(1,\dfrac{1}{2}\right)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2}+3y^{2}=6$ on any tangent to it is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 3 JEE MAIN PYQ

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity ${1 \over 3}$ and the distance of the nearer focus from this directrix is ${8 \over {\sqrt {53} }}$, then the equation of the other directrix can be :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1$ at the points $A$ and $B$. Let the line $y=x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 5 JEE MAIN PYQ

The centre of a circle $C$ is at the centre of the ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$. Let $C$ pass through the foci $F_{1}$ and $F_{2}$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $PF_{1}F_{2}$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

An ellipse, with foci at $(0, 2)$ and $(0, -2)$ and minor axis of length $4$, passes through which of the following points?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

Let the product of the focal distances of the point $\left( \sqrt{3}, \dfrac{1}{2} \right)$ on the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $(a > b)$, be $\dfrac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

The length of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose midpoint is $\left(1,\dfrac{9}{5}\right)$, is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

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 \[ 25\big(\beta^2 x^2 + \alpha^2 y^2\big)=\alpha^2\beta^2 \] is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

Let the length of the latus rectum of an ellipse with its major axis along the $x$-axis and centre at the origin be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

Let the length of a latus rectum of an ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ be $10$. If its eccentricity is the minimum value of $f(t)=t^{2}+t+\dfrac{11}{12}$, $t\in\mathbb{R}$, then $a^{2}+b^{2}$ is equal to:

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

Solution

Qus : 12 JEE MAIN PYQ

Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, 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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

Let $e_1$ be the eccentricity of the hyperbola $\dfrac{x^{2}}{16}-\dfrac{y^{2}}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ ($a>b$), which passes through the foci of the hyperbola. If $e_1e_2=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

The equation of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose mid-point is $(3,1)$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

If the curve x² + 2y² = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :

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

Solution

Qus : 16 JEE MAIN PYQ

Two sets $A$ and $B$ are as under : $A = {(a,b) \in \mathbb{R} \times \mathbb{R} : |a-5| < 1 \text{ and } |b-5| < 1}$ $B = {(a,b) \in \mathbb{R} \times \mathbb{R} : 4(a-6)^{2} + 9(b-5)^{2} \le 36}$ Then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (Offline) PYQ

Solution

Qus : 17 JEE MAIN PYQ

If x² + 9y² $-$ 4x + 3 = 0, x, y $\in$ R, then x and y respectively lie in the intervals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 August Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

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

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

Solution

Qus : 19 JEE MAIN PYQ

If the midpoint of a chord of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is $\left(\sqrt{2},,\dfrac{4}{3}\right)$, and the length of the chord is $\dfrac{2\sqrt{\alpha}}{3}$, then $\alpha$ is:

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

Solution

Qus : 20 JEE MAIN PYQ

If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus an its nearest vertex on the major axis is $\dfrac{3}{2}$ units, then its eccentricity is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

Let $S$ and $S'$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\triangle S'BS$ is a right-angled triangle with right angle at $B$ and area $(\triangle S'BS)=8$ sq. units, then the length of a latus rectum of the ellipse is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on the ellipse. Then $6f_1f_2 - r$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

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

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

Solution

Qus : 24 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $\triangle AOP$ is $4$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

For some $\theta \in \left( {0,{\pi \over 2}} \right)$, if the eccentricity of the
hyperbola, x²–y²sec²$\theta $ = 10 is$\sqrt 5 $ times the
eccentricity of the ellipse, x²sec²$\theta $ + y² = 5, thenthe length of the latus rectum of the ellipse, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 27 JEE MAIN PYQ

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

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

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

Solution

Qus : 29 JEE MAIN PYQ

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $\left(0,5\sqrt{3}\right)$, then the length of its latus rectum is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\dfrac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices at the vertices of the ellipse, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

Let $A(\alpha,0)$ and $B(0,\beta)$ be points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E:\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and let the corresponding focus be $S$. If the perpendicular from $S$ to the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

The eccentricity of an ellipse having centre at the origin, axes along the coordinate axes, and passing through the points $(4,-1)$ and $(-2,2)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

Let $f(x) = x^{2} + 9$, $g(x) = \dfrac{x}{x - 9}$, and $a = f \circ g(10)$, $b = g \circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\dfrac{x^{2}}{a} + \dfrac{y^{2}}{b} = 1$, then $8e^{2} + l^{2}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Morning Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

If $S$ and $S'$ are the foci of the ellipse $\dfrac{x^2}{18} + \dfrac{y^2}{9} = 1$ and $P$ be a point on the ellipse, then $\min(SP \cdot S'P) + \max(SP \cdot S'P)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

The area (in square units) of the region enclosed by the ellipse $x^{2}+3y^{2}=18$ in the first quadrant below the line $y=x$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

If the length of the minor axis of an ellipse is equal to one-fourth of the distance between the foci, then the eccentricity of the ellipse is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

Let ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function,
$\phi \left( t \right) = {5 \over {12}} + t - {t^2}$, then a² + b² is equal to :

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

Solution

Qus : 39 JEE MAIN PYQ

If the co-ordinates of two points A and B are $\left( {\sqrt 7 ,0} \right)$ and $\left( { - \sqrt 7 ,0} \right)$ respectively and P is anypoint on the conic, 9x² + 16y² = 144, then PA + PB is equal to :

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

Solution

Qus : 40 JEE MAIN PYQ

Let $C$ be the circle of minimum area enclosing the ellipse $E:\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ with eccentricity $\dfrac12$ and foci $(\pm 2,0)$. Let $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $PQR$ is:

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

Solution

Qus : 41 JEE MAIN PYQ

Let ${E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b$. Let E₂ be another ellipse such that it touches the end points of major axis of E₁ and the foci of E₂ are the end points of minor axis of E₁. If E₁ and E₂ have same eccentricities, then its value is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

The length of the latus–rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\dfrac{4}{5}$, is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 44 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 July Morning Shift) PYQ

Solution

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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Ellipse PYQ

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